a survey of the infrared radiation properties of carbon...
TRANSCRIPT
C.P. No. 981
MINISTRY OF TECHNOLOGY
AERONAUTICAL RESEARCH COUNCIL
CURRENT PAPERS
A Survey of the
Infrared Radiation Properties
of Carbon Dioxide
BY
J. P. Hodgson
LONDON: HER MAJESTY’S STATIONERY OFFICE
1968
Price 15s. 6d. net
C.P. No. 981*
December, 1966
A SWWY of the infrared radiation pronerties of carbon dioxide
.I. P. lndpson Department of the Nechanics of Fluids
University of :lanchcster
This paper consists or a survey of theoretical and exncrimcntal approaches to the prediction of the low resolution emissivitv of an infrared bqnd of a polyatomic molcculc. Cpcctrnl band models nnd molecular nodels are discussed, with carbon dioxide beinr! tahcn as an examnle of such a noleculc.
____________________----------------------------------------------------------
‘Qeplaces ~AA:B.C:ZB 645 r
-2-
List of Contents
Section l(a)
lb)
l(c)
2(a)
2 (b)
364
3(b)
4(a)
‘i(b)
4(c)
5
6
7
Introduction
Einstein coefficients in a system of two- level particles . . . . . . . . . . . . . . . . . . . . ..a..........
Radiation density, emissivity and integrated absorption . . . . . . . . . . . . . . . . . . . . . . ..*............
Quantum mechanical method for determining the Einstein coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .
Matrix elements and selection rules for diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...*..
The carbon dioxide molecule and its infrared spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The result of an application of the line sum- mation method to the calculation of the total relative band intensity of the carbon dioxide 4.3~ band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A special method of calculating the partition function of carbon dioxide in application to the 4.3~ band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radiation laws and definitions . . . . . . . . . . . . . . . . .
The equivalent line widths of spectral lines for various line shape parameters . . . . . . . . . . . . . .
Models of band spectra and calculation of spectral emissivity in terms of equivalent widths of single lines . . . . . . . . . . . . . . . . . . . . . . . . .
Application of the statistical model to the 4.3~ band of carbon dioxide and calculation of the spectral emissivity . . . . . . . . . . . . . . . . . . . . . . . .
Direct application of the statistical model at 1200°K, in order to verify its validity, and CO derive an expression for the spectral absorption
Conclusion and a brief discussion of spectral and molecular models . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction
Page
2
4
5
8
14
18
26
27
34
40
43
48
56
61
Like solids, gases can emit, absorb and transmit radiation according to the elementary laws of radiation. The emission or absorption spectrum of
-3-
a gas is not continuous but 1s distributed into bands containlnE many sinp,le spectral lines caused by well-defined quantum number chances within the Kc36 molecule.
Except III the presence of external electric or magnetic fields there are three types of quantised transitions to be considered. These are:
1. rotational transitions 2. vibrational transitions, and 3. electronic transltions.
For a typical molecule the change of internal energy AE, for the lndivulual transltion increases down this list. That is,
AE1 << AC2 << AI: 3
Thus one particular type of transition dominates the energy change III a general transition in which several quantum number changes are involved, and the bands are named accordingly:
1. rc~ ational bands, typically in the wavelength range 10 5 to lo4 microns,
2. vibrational bands, typically in the wavelength ranye 1 to 30 microns, and
3. electronic bands, 10-2
typically in the wavelen@ range to 1 micron.
It should be noted that electronic bands contain an electronic transitlo”, which defines the spectral position of the band, and also vibrational and rotational transitions which cause a large number of different spectral lines to be formed close to the positions defzncd by the electronic transition. Similarly for a vibrational band, a chany,e in vibrational quantum number defines the spectral position whereas the actual lines of the band are caused by sinultaneous vibrational and rotational transitions. Fo; this rcnson vibrational bands are often referred to as “vibration-totntion” bands.
The actual position of a band of course depends entirely on the size, shape and internal forces of the molecule concerned.
Here we are concerned entlrely with vibration-rotation bands under low spectroscopic resolution. For any given polyatomic rrolecule there are many vibration-rotation bands, each one accompanied by a particular chanv in vibrational quantum number(s). Under low resolution the rotational lines of the band are not distinguished and the overall effect of these lines is to broaden the band.
The spectral properties of the carbon dioxide molecule are reviewed top;etber with the calculation of the integrated band intensity of absorption and the spectral emissivity. Radiation laws and some useful parameters are defined and a survey of spectral band models is Five”. The 4.3 micron band of carbon dioxide is the band of Rreatest interest here, but the arguments can be extended to apply to other bands of the carbon dioxide molecule and to bands of other molecules.
-4-
lb) Einstein Coefficients in a System of Two-Level Particles
The basis of the method involves the solution of the time- dependent Schrodinger equation in the presence of a perturbation potential.
To understand the physics of the problem let us consider a simple system of molecules each of which can be in either of two states, with energies Eu and Et, where Eu>Et.
The respective populations of these states, N and Ne, at a temperature T are related by the Boltsmann distributiox function so that
(1.1)
where g constad.
and g are the degeneracies of the states and ‘k’ is Boltsmann’s This can be rewritten
(1.2)
where E - E u 9. = hv ue defines vut. These equations describe the equilibrium
population of the system, but if for some reason the equilibrium is disturbed (for example by a sudden decrease in surrounding temperature), the above equations no longer apply and there will be a redistribution of population, restoring equilibrium. (In the particular example quoted, molecules with energy E will decay by radiation of a photon into state E until the above eqxations hold again at the new temperature. Molecu ar collisions f are not considered, internal dynamic equilibrium being re-established by radiative transfer only).
We now consider the above system to be enclosed in unit volume and subject to a spectral radiation field of density p until the whole system has reached equilibrium. There will be three m”Bin processes going on, each one characterised by an Einstein Coefficient(l).
Spontaneous Emission
The higher energy molecules decay so that the number of transitions is given by
Nu Aut per cm3 per sec.
where A us is the Einstein Coefficient for spontaneous emission.
Induced Emission and Absorption
The molecules are also subject to interactions with the radiation field and a collision with a photon of energy (E - Et) can induce a change of state in either direction. such that tge number of upward transitions is given by
where B eu is the Einstein Coefficient for absorption, and B ue is the
Einstein Coefficient for induced emission. Usually we have Ne>>N ll
and absorption dominates induced emission, but if No>>N 9. the situation
can be used for light amplification (LASER) provided AUp is not too large.
Since the system is in equilibrium
N” A”f, = Ne Bd’“u9. - NU B p ue we
and it follows that
(1.3)
(1.4)
h hvuQ Pm - R, kT
For black body radiation systems in equilibrium at the temperature T,
P yue _ Bnhvo; h”“Q
c3 (e - kT - 11-l
and it can be seen, by solving the second part of (1.4) for oLIup
and comparing with (1.5) that the Einstein Coefficients are inter-related. In fact
% Bh = gu BUQ 8nhv3ue
, A =3 ue B
ue . c
6) Radiation Density, Emissivity and Integrated Absorption
Radiation density o,(T) is defined as the radiant energy per unit frequency interval at frequency v. The velocity of propagation is the velocity of light ‘c’ and by using the arguments of kinetic theory we can show that the isotropic radiant energy, incident in time dt, on area of wall dA, in frequency interval v to v + dv is
t c o,,(T) dvdA dt.
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It follows from Kirchhoff’s Law that the radiant energy emitted from area dA of a non-black body at a specified temperature in time dt into a solid angle of 2n and in frequency range v to v + dv is
EARS’ dAdtdv = $ c p”(T) dAdtdu, (1.6)
where R{‘(T) is the Pl as the
anck blackbody function (15) and c”(T) is defined emispherical spectral emissivity.
The effect of absorption by a sample of gas on which radiation is incident can be described for a collimated beam by the spectral absorption coefficient per unit length in the gas, K,,v which is defined by the relation
*
- dpvdv = P, %,“dL d”, (1.7)
which states that the net decrease in radiation density is proportional to the incident radiation density and the path length ‘L’ of the radiation in the gas. The constant of proportionality is the spectral absorption coefficient. The path length dL is traversed in time (dL/c), therefore for an incident radiation density of p”(T) the radiation energy lost per unit time is _ c dpv dv , and so the number of induced transitions
dL from energy level Ef to level Eu , due to radiation in.the range v to v + dv, is
-k dpv dv per unit time.
Therefore the total number of transitions‘ per unit time is given by
N -I ‘= A”
“” tt,” dv ,
“9. hv
where A” ua indicates integration over the line width at frequency v “a’ If the spectral line is very narrow and pv does not change much across it then
CP N “UP.
tr =huuQ i; ““ue
%,“d” (1.9)
Induced emission is coherent with the incident radiation, and this expression can be equated to the difference in numbers of induced transitions u to L and e to u. In terms of Einstein Coefficients this gives
N tr = We Blu - N" B"a)P"e"-
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Absorption coefficients of rases under normal circumstances are proportional to particle densitv and so we define the interrated line absorntlon (1) as the integral of the spectral absorption divided by pressure ‘p’ and wave velocitv ‘c’
i.e. 1 St” = - PC s 5,,” dv
A”“k
we see from (1.9) and (1.10) and (1.2) and(l.5)
1 hv up. N %u = 7 * CPyUe tr
= hvue
PC2 CNe Be” - N” B”,)
= hv”Q N”aUt - I~eBeU (1 -r 1 PC2 e au
= hvue h”“k -NB PC2
e e” (1 - e -FT),
hvUe h”uQ i.e.
-- S 9.u =2 liQEQu (1-e kT 1,
PC
(1.11)
(1.12)
SQu is usually measured in units of (atmospheres) -1 x (centimetres) -2 .
The value of the integrated line absorption gives us a measure of the brightness of a spectral line, and is a function of temperature.
The statement that absorption coefficients are proportional to pressure is not quite true since they are really proportional to density and only proportional to pressure at a fixed temperature. This method of standardization is common in gas dynamics, and it would be better to use the standard density unit, the amagat, except that we normally measure pressures rather than densities. In what follows, the units for optical path length will be taken as pressure times geometric path len th,
-f and
the units for integrated line intensity will be (atmospheres) (centimetres) -2
as stated above. This step is taken only to fall in line with previous work, which makes data correlation easier, although it would be physically more correct to say
-8-
S 1
eu ‘G s K
L,v dv (amagatrl an*-*
Av”k
The picture so far enables us to calculate the line intensity for a transition provided that we can calculate the Einstein Coefficients. The quantity St is independent of the line shape which is given by
%v- We shaly find that when applied to carbon dioxide the suffices
1 And u embrace several quantum numbers, since the upper and lower states have degeneracies. We shall then sum Stu over appropriate
quantum numbers to give us the integrated band absorption.
We now give: an abbreviated account of the quantum mechanical method of determining the Einstein Coefficients.
1 Cc) Quantum Mechanical Method for Determining the Einstein Coefficients
This is a perturbation method (?,3), the validity of which is based upon the assumption that the radiation Hamiltonian has only a small effect on the unperturbed state of the system.
The Schrodinger equation may be written
H$ = a*
Ifi -z ’ (1.13)
where H is the total Hamiltonian operator. We can represent H by H
= Ho + H’ , where Ho is the equilibrium Hamiltonian operator and H’
the perturbed Hamiltonian operator. If H’ is zero the solutions of the equation
(1.14)
are $ = $n” , and $,’ represents a complete set of wave functions which
can be expressed as
iE t
*no (9, t) = bno (2) e- +- ,
where E is the energy of the state 6,‘. becomes”
For H’ non-zero the equation
a* (H’+H’)o = ih =, (1.15)
and to obtain a solution we write JI - f; cn(t)JInO(qrt) and substitute
-9-
into the equation above civinp:
I: H”cn(t)O ,,“(Q) n + fi PC,,(t) W,,?J, f)
= i* I: fin *no
a* ’ n dt (a, t) + ltr i c,(t) -” at (1.16)
But (1.14) tells us that the first and last terms here are equal, so that
i H'+,(t) $J,"(cJ, t) = i-h g $ vno (9, t). (1.17)
Icp I ’ is the fractional concentration of $,“(J, t) in the state $(q, t).
To obtain the rate of change of concentration of each state we
multiply (1.17) by JI ‘* , m
and integrate over all space,
i.e. i* I $ 0* m E% JI’dT ndt n ; H’c$) Jl,‘dT
Since *no 0*
represents a complete set I Jim qno d? = 6mn. Therefore we
have left
dc c z 1 2 n3l
cn / em’* HVno dT
(1.18)
(1.19)
This last eouation has no physical application as it stands, but the interral on the right is related to the probability of a change from state m to state n due to the presence of the perturbation Hamiltonian.
It can be shown that for a radiation field the Eamiltonian H’ has rhe form (1)
where e. is the charge of the jth particle, with mass mj and A is the vector rhagnetic potential of the radiation.
First consider a light wave polarised in the x-direction and
- 10 -
travelling in the z-direction; for this wave A2 = A., = 0 and
A1 = Al’ CO6 2nv (t -
where A 0 . 1 1s a constant.
Substituting the expression for Hl ’ into the integral of equation (1.19) we get
d Al,j ;T;; *n ’ dr
j
For visible and infrared radiation the wavelength is very much greater than the molecular dimensions, so that A will be a constant in the region of the molecule, i.e. A - A 1.j
1.j 1 and then
I I&‘* H1’ (I,’ dT = xA1 e - i(E, - E,)t
c 47
x ; 2 I $ ‘* J In
&. JI,’ dl (1.21) j
If we consider the time-independent wave equations for the two states
*no nl” and $ we can show that
j enodT = - %E -En)I$‘*xj *nod7 *2m m (1.22)
so that (1.21) reduces to
- E I J,‘*Hlt
*)t
m ,&‘,-jT = - &
i(E,
A1 (Em - En) e 4 jxmn (1.23) ll
where
X Inn Jl,,‘dT (1.24)
That is, through the transformation (1.22) the integral in (1.19) becomes a
- 11 -
function of X , the square of which represents the x-component of the ~lecular dipEye moment matrix element (4) for a transition between the two states m and n. (1.19) reduces to
dc 41 1 i(E, - E,)t
YP=- &2 n y,(E, - En) e- li Xm”* (1.25)
for the physical system in which we have a fractional concentration of the mth state at t = 0 of c,(O) = 6mn. For perturbations from the initial state it follows that
i(E m - E,)t
dc *l X,, (E JP = -2 Ill - En) e- tr
of Also for 1iehtAfrequency v at z = 0, Al becomes
(1.26)
Al = Al’ b 2rivt + e -2nivt
1
whence,
dc P’- Al0 X -
2ck2 mn (Em - En)
Integrate from
C m -Gi
i(E, - En + hv)t i(E, - En - hv)t x tl + e il 1 (1.27)
(t=lJtot=t
Icm = 0 to c m = c,(t), then
AloX,,& - En)
, 1 i (Em - En + hv)t i(E, - C - hv)t
x e tr k” -1 (1.28)
Enl - En +hv
-1 +e Em-E -hv n I
which is large only for E - E - + is of order unity.
hv, in all other cases the expression in brackets If we?ake” E>En, then we can have only Em - En - +hv
and the first term is never large.
The probability that the system will be in a state m after a time t
- 12 -
is therefore
t2 i- 4A4
IA;1 2 b$,,,12 (Em - Enj2
m s
CEm - En - h )t sin* 2-h 1 dv x (Em - En - hvJ2t2/&k2
0 (1.29)
This leads to
lA&,,J I2 IX,, 12, t (1.30)
So far we have considered light fully polarised in the x-direction. For randomly polarised light, the cross terms vanish and we have left
lT2v= lcm(t)12 = 3 [ ~l~2~m~2 + lA2’t2 &,I2 + 1A3012 \zmn12]
(1.31)
where 1~~1 2 and 1 zmn( 2 are the y, z components of the electric dipole matrix elements and A2’, A3’ the respective constant amplitudes of the incident radiation. so that
For isotropic radiation 1~1’1 2 = ~~~~~~ u 1~3’1~ = fig *,
JcJt)12 n2vli a hl”“l’ lRmn12 t. (1.32) where ~~~~~~ = IXJ + Iymn12 + bm12 is the square of the modulus of the total electric dipole matrix element
IR,,I 2 = s
orno* T. ejAj $,,Odr 2 j
(1.33)
We can relate (_A“o(u~)~~ to the radiation density, since
- 13 -
A = A0 - - sin (2nvmnt)
2A and E 5 -1 -
c 37’. where 2 is the electric field,
2nv a-2 A0 co6 (2nvmnt); c -
therefore (:12=
= 4nrvmn.
nv2 therefore P mn =-!I!! lg12,
2c2
so that 1 c,,, (01’ = E2 p,,,,” IRm12t.
Now Ic,(t)[’ /t is the probability per unit time that a molecule of the
system will undergo a transition to state m from state n under the influence of the radiation density pvmn , where E,>E
n- It follows from (1.3) that the
number of upward transitions par unit time is
N” = N” BnnJ P~“ln *
Therefore
Bm ~I~ m”
-5 km I2
64n4v3 and A= - 6”
In” 3h c3 x lR,,,“12 .
(1.34)
Hence we have succeeded in relating the Einstein Coefficients to the sum of the squares of the matrix elements of the dipole moment components which can be evaluated for any physical system if ALL the wave functions are known. This last condition is the overriding G.
So far we have shown that the integrated line absorption due to a
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transition from a state with quantum number n to a state with quantum number m is given by
8x3” hv
s = nm 31Tc2
(1.35)
The wave functions and selection rules for diatomic molecules will now be briefly discussed, and developed later to apply to the carbon dioxide molecule.
2(a) Matrix Elements and Selection Rules for Diatomic Molecules
For a diatomic molecule the Schrodinger time-independent equation can be solved in the harmonic approximation (1.5). The resulting wave functions govern the transitions that can take place when a molecule is subjected to a perturbing radiation field. This leads to selection rules. We consider now the one-dimensional harmonic oscillator.
The complete time-independent wave functions for the one-dimensional harmonic oscillator are the orthonormal Hermite polynomials .
[ 1
1 @X2
Jl”(X) = ~ (2)1 1
2”“: H,(& x)e - -?
where x = r - r distance and CL
e is the displacement from the equilibrium internuclear is related to the potential energy ‘V’ through
n2x2h2 V(x) - 2m t
r
where m given b$
is the reduced mass of the molecule. The energy levels are
E” = (“+h)hvo.
In the dipole approximation we can represent the electric moment,u by
u = ; + y(r - re) - g + fx,
(2.1)
where g is the permanent dipole moment of the molecule, and u is induced 1 by the departure from equilibrium. If we now calculate the matrix element
xm” for a change of state due to a perturbing radiation field, we find that
- 15 -
c = “,” I:
producing. no chanp,e of state,
S” *+I = ,
X” “-1 = ,
and
(2.24
(2.2b)
For all other values of m # n, n + 1 we find X “Ill = 0, leadine us to the
selection rule in the harmonic approximation of An = + 1. It is the
value of !J 1 which we have difficulty m findins and the point at which we
need further empirical information. Later (Section 3) it ml1 be seen that we can solve the problem fairly satisfactorily.
The method now adopted is to proceed to find the matrix elements and the line intensities, which we expect to contain constant factors of proportionality related to the magnitudes of the dipole moments. lie consider the vibrational levels to be degenerate due to the total rotational increrrents of angular momentum and also due to the orientation of this angular momentum in space. The former give rise to the rotational quantum number K which can have non-neaative intezral values and the latter give rise to the maE:netic quantum number M, which can have (2K + 1) values extendins fron -K to +K.
We consider the integrated line absorption (defined in (1.12) and also called the ‘line intensity’ or ‘line strength’)due to a chance of state from (n K M) to (n’ K’ M’) (see paqe 8) where the changes in
/
quantum numbers are governed by the selection rules of the harmonic oscillator and rigid rotator.
Thus IRn,I’ becomes lRnKM ~ n,K’M’12 .
The selection rules limiting K and M are, for
x,y polarized lieht AM = 21, AK = + 1,
z polarized lip.ht, AM = 0, AK = + 1.
Using this inforrration we can sum over all possible transitions for a vibration quantum number chanR:e of n to II’, and hence calculate lRn,,1*
=*d o’“,,I , the vibrational band intensity. This method, due to Herman (6.7)
is now described briefly.
- 16 -
Let the jth component of the matrix element given in (1.33) be
where I(r, 8, ?,) is the jth component of the electric dipole moment
vector Il(r, 8, 5). *nm is the complete time-independent wave function
of the diatomic molecule, which we can separate into the form
$ nl@l = SnK(‘) gm(8* c)r
and we can also write y cr. er 5) - v(r) Fj(O, 5). We can now sum
IRm.+n-K~M' 1' war all transitions in which the selection rules are
obeyed. For the surmnation over M M' and components j we have
and it can be shown (11) that the summation on the right is given by
W6 K-l, K’ + W+l) 6K+1,K,),
where& is the Kronecker h-function.
Corresponding to the matrix element IR-~,~,~,~~ we have an integrated line absorption
s nKM+n'K'M' = & Nm
31ic2 p "(nKM, n’K’M’)
hvbKM, n'K'M') x 1-e-w
I 1 RnKM+n'K'M'I 2 (2.4)
- 17 -
In the absence of magnetic fields the levels are degenerate due to the inability of the mapnetic quantum number to split the levels, and this depeneracy IS gnK(= (ZK+l) for the state n, K) and so
N N nK
“KM = 1 nk -
Therefore
an3 N”K s”K+“‘K’ - - --
3.h r V(“&n’K’)2&1
h” (“Kc, n’K’) x (1 - e-i7 ) x 1 J SiK u(r) Sn,,<,d+
,:el K + (K+1)6K+1 K;’ . 9 , 1 (2.5)
The values of the integral can be calculated in the harmonic * . approxmatmn but contain the unknown factor due to the magnitude of the dipole moment vector; of this experimental knowledge is indeed very limited.
By a similar summation over KK’ we can estimate the total vibrational band intensityW. However, this is more difficult than the summation over M since there is a variation in frequency over the band width. The result is
an3 Nn - P- 3tlc2 T "n,n' lRna,12(l - e - %I), (2.6)
where 7 Owing nD”’
is a complicated function of the average frequency in the band, to the approximation of complete harmonicitv the values of
“9 n’ are limited to being small and so the formulae can only be expected to hold at low temperatures and even then we can only find the relative band strength. Thus even if we know the band strength at room temperature, it is difficult to get a reasonable value for the band strength at much higher temperatures, since the harmonic approximation is no longer valid.
Before applying the above method to the carbon dioxide molecule it is necessary to know more about this molecule and its modes of vibration. The method (given in Section5) will be that formulated by Malkmus, which assumes a band intensity at room temperature and an average line width, with a given temperature variation.
- 18 -
2 (b) The Carbon Dioxide Molecule and its Infrared Spectrum (5,9)
The molecule has three atoms and therefore nine degrees of freedom. We can divide these into internal motion and external motion of the molecule as a whole.
(1) The external motion consists of translation of the molecule as a whole in the three orthogonal directions and, since the molecule is linear, the rotation of the molecule about the two axes perpendicular to the axis of the molecule.
(2) The internal motion consists of three modes of vibration and one of rotation which only becomes effective when one of the vibrational modes is excited.
By a linear transformation we can reduce the internal motion of the molecule to its normal co-ordinates, (x, y, z, q).
The kinetic energy is then given by
T = 1 rn4’ + fu(i2 + 9’ + i2)
where q=(z 2 - 21) - = , x=x 3 - 4(x, * x2),
Y’Y 3 - ICY1 + Y,), 2 = 23 - lb, + z2)*
r = x2 2 2 +Y *x - r co6 $, y = r sin 4,
2Mm and ” M+Zm
This reduction to normal co-ordinates depends on the assumption that the vibrational amplitude is very much smaller than the inter-atomic separation.
To the harmonic approximation the potential energy is
(2.7)
- 19 -
v - v” 2 22 .‘ll mu1 q +2n2”2v(x2+y2) +2*2v2 2 2 3 PZ . (2.8)
The solution of Schrodinger’s equation for this potential leads us to the vibrational wave function
“1”2”3’ “1 * - J, (O)$
“3tsj R”2:p) e? ie* (2.9)
where the non-dimensional quantities (I, 5, p a q, z, r respectively,
tinl(o) and $1~3 “zP. (5) are orthogonal Hermitian functions and R (P) =
Pee - 3-12 ” -f. 2
E W-O
aKpK :
Also, the total internal energy of the molecule (to the harmonic approximation) is
Wnl”2”3 - (“, + 1, hvl + (“, + 1) hv2 + (n, + i)hv3, (2.10)
so there ife thiee fundariental vibration bands in which each of (n,, n . 2’ “3) changes by one unit.
The fundamental modes of vibration and the significance of the quantum numbers are demonstrated below (Table 1). The vibrational tem- peratures, frequencies and wave lengths are also given (10).
It can be see” that in the symmetric stretching mode there will be no change in dipole moment of the molecule, so that the matrix elements are all zero. This band’@ not observed in the infrared spectrum of carbon dioxide but only in the Raman spectrum. It does however combine with the other two modes giving rise to combination bands in the spectrum.
r wave- wavs- hv Type of Quantum ‘length
(ld number 0 -r
-1 ,,y mode. number:
cm OK
: 4 4 15 667 ,959 bending * z * “2 7’5 1336 1920 symmetric
stretching +. . .* “1
4.3 2349 3350 asymmatric stretching .+ C. .* “3
- 20 -
The quantun number '4'. associated with the bending mode, is a measure of the angular momentum about an axis parallel to the ground state axis of the molecule. Thus 'C' must be zero whenever' the bending mode is unexcited, and it is found that the values that '&I can take ate
"2, (n, - 21, (n, - 4) . . . . . 1 or 0. This mode of vibration introduces a"
asyrmnetry into the molecule about the Ground state axis of the molecule, so that the dipole lpoment matrix element is "on-zero and furthermore the change of dipole moment is perpendicular to the axis of the molecule. This characterises the type of band spectra emitted by chan$+es of quantum number n 2' Only such bands are observed for which (9)
An2 is ODD, AC = + 1 and A" 3 is EVEN.
The asynnnetric stretching mode has asymmetry parallel to the axis of the molecule so that a chawe of nuantum number n 3 causes a dipole moment change parallel to the molecular axis, which produces a characteristic "parallel band" spectrum. Only such bands are found for which
A"2 is EVEN, AP. = 0, An 3 is ODD.
To avoid confusion as to the values that An2 and An3 can h,fve, it is as well to note that in the harmonic approximation‘the selection rules demand that An =+l, 2 - An =O 3 for a perpendicular band and
A"2 = 0, An3 =L 1 for a parallel band. All other ouantal jumps are forbidden, but ONLY in the harmonic approximation. In practice'the potential energy wells of the molecule contain much anharmonicity and this increases the possibility of quanta1 steps ex&edinp the 'harmonic restriction ofAn = + 1, especially for large values of n. HoweVer, it should be pointed out that the probability of a &an&in quantum numbers rapidly decreases with increasinp magnitudes of the change. Thus we "' can readily accept a violation of the selection rule An = + 1, thou+ the band spectra produced by changes of An> 1 are of a muc% weaker intensity. Therefore in the above restrictions on An ,and An transitions read zero for even and one fo? odd.
3, for the most probable
The selection rules rovernin- chanpes in ‘&’ are much score rigorous since the ripid rotator node1 is a better approxirration to rqtational motion than the harmonic approxination is to vibrational motion.
The above discussion can be verified by applyinq a small pertur- bation to the harmonic potential and calculatlnr the transition probabilities from the resulting perturbed wave functions. The two statements on An value of A" 1 is 2' Ae and An3 are the result of this work (9) and the
show" to be of no significance here.
To realize what these results mea" consider the spectral bands that appear at the frequencies
I\"1 "1 + An2 u2 + A"3 v3 (An's are of course integral).
Onlv such band5 are found for which (An2 + A”3) is an ODD nurr.ber, as we have see”.
(9 For An odd dlrole nolecu ar axis , i
(ii) For An odd dipole molecr1 ? ar axis.
These restrictions
moment chnnoe is perpendicular to the
moment chnn:e is parallel to the
have an important effect on the spectrum
- 21 -
of the carbon dioxide mdlecule, I” particular in the formation of combination hands. Consider the transitions leading to the two observed hands:
1. Y’ = An;vl + A”; “2 + I\n;v3
2. Y” = A”% 11
+ An-J ? 2
+ A”% 3 3
A combination band would be
(2.11)
(2.12)
3. (v’ + Y”) = (A”; + A”‘;)u~ + (An; + A”“)v 2 2
+ (A”’ + A”“)V 3 3 3
(2.13)
Since bands 1 and 2 are observed both (An’ 2
+ An;) and (A$ + An;)
must he odd; therefore (An; + An”) + (An’ + A”; ) must be even, so that 2 3
a combination band of two observed hands cannot he observed. 1:owever ) the eymmetrx node often interacts with the other two modes producing observed bands, since An 1 is unlimited; for example,
“=” l+” 2 oo”0 ++ 1110
u=Y +u 1 3 oo”0 - lo”1
We have see” that we can predict the positions of the various bands knowing the three fundamental frequencies, and we shall now investigate the structure of the bands which ~111 be found to be of two main types (5). In order to do this we shall need to consider the rotational motion more closely.
By choosing axes (5, ‘1.5 ) to b e along the principal axes of the molecule we can ensure t’hat the Hamiltonian will depend only upon the principal moments of inertia (A, B, C ). Let A,B be the two equal moments of inertia and let the linear molecular axis lie along t with moment of inertia C. (The following: analysis applies to any triatomic molecule).
- 22 -
The orientation may be described by the three Eulerian angles (0, $,?c). After forming the Hamiltonian and Schrodinger’s equation for V Z 0, we can separate variables and solve for the wave functions
Y= 0 (cl) e iK$ .iM55
where K and M are integers, in order that ‘4 be single-valued. From the resulting equation for @ (0) we can show that the energy levels are defined by
J(J + GIJK - 2A
1) (2. ,14)
where J is a positive integer. The relationships, given by the equation, between J, M and K suggest their physical significance. These relations are
J 2 (KI , if IKI ‘, [Ml ,
J b IM( , if (Ml a IKI ,
i.e. J. is greater than or equal to the larger of the magnitudes of the two quantum numbers K and M. On the other hand we may say that for fixed values of 3, there is a limit on the values of 11 and K, such that
IKl EJ and (Ml f J .
This suggests that J, K, M can be interpreted as representing angular momentum quantum numbers, and it can be shown that angular momentum is given as follows:
- 23 -
the total annular momentum L is civen by L = J(J + 1)312 * the component along the r, axis by LC - k3 ,
the component along the z axis by La P Mh .
From the formulae restrictine certain values of M,K it can be shown that the degeneracies of the states of angular momentum (J,K) are given by
'JK = 23 + 1 K = 0,
- 2(2J + 1) K > 0, etc.
Selection rules can be found by determining the matrix elements representing the ciirection cosines p... The results are the rules
1.l J =+ I.0 ; K=O; M = + 1, 0, for which p
5x' pcY and o r,7.
have matrix elements differing from zero, and AJ = + 1,0 ; AK = + 1; AM = + 1,0 for which p value< of matrix eyemen ex ' Orly ' pnz and p
9. cx, pcy , pcz have fin?'te
Assuming: the above expressions for the degeneracy and that the total intensity of radiation from a state is proportional to the degeneracy of that stat&, we can find the relative intensity of the rotational lines in the vibiktional band.
The above two sets of selection rules lead to two types of band, called parallkl and perpendicular according to the direction in which the change of dipole moment occurs with respect to the molecular axis. This corresponds to the change in quantum number K, which can be 0 or + 1.
(9 Selection rules AJ '2 1, 0 and AK = 0,
Refekence to the energy constant wJK for this molecule shows that the lines of &I absorption band will be given by the expressions in the following table, where Y is the normal frequency of vibration. The intensities of thk lines'in the three branches corresponding to AJ = -1, 0, +l, are also given (5)
- ve branch: J,K tlJ IJ "J-l = "o - ?;;A ' J-l
D AJ;l Q @ ea(J2+J)-~aK2
K *a 0
(2.15)
J 0 branch: v$ - v. ; I 0 J( ? I: K2 2J+l - o(~2+~)-go K2
J-1 KPO 23ziYe (2.16)
+ ve branch: v J-l,K W J vO +%i ; IJ-1
-o(J2-J)-t30K2 (2,17)
J,K -
- 24 -
where & is a constant dependinp on the molecular dipole Foment matrix elements; 9 = 1.2 where K = 0 or # 0, lT2 A CT== and B-?-l.
So far this discussion of the parallel hand has been for a general triatomic rrolecule, but for a linear molecule, such as carbon dioxide, vibrating in the asymmetric stretching mode, the rotational motion is like that of a symmetric rotator. The moment of inertia C tends to zero, so t)lat 6 becomes infinite. However, since C is so near to zero the quantum n~nber K cannot have any meaning, except when it is zero. The immediate consequences of this is that the zero branch vanishes and the intensities are represented by the much simplified expressions:
IJ 2
J-1 = ! J J ,-a(J +J)
IJ-1 e J
! A iJ g'l(J2-J)
which are seen to be very similar to the intensity variations of the vibration-rotation hand of a diatomic molecule. The intensity variation is a direct function of o hut the form of the band depends little on its VdUF?. Since, when L is non-zero, the molecule loses some of its symmetry, we have twice as many lines in a band for P>O, as well as a weak central branch. In fact in the hand for $! = 0, these lines are superposed in uairs, the lines with odd values of J being absent. The moment of inertia C becomes finite as in a general triatomic moleCtile,if L>O. Fig. 1 should help to clarify this point.
(ii) Selection rules AJ = + 1.0 and AK = 0.
Again reference to the energy level constant W .lK will rive us the line structure of a general triatomic perpendicular band, which is more complicated than the parallel band. The general appearance depends preatly on the ratio (A/C), which is infinlte for CO in a state with e = 0, but it is useful to look at the case for whizh e>O, i.e. (A/C) finite. This band can be described by a series of superimposed single bands, two sets of which are given for values of AK = + 1, and each single band has positive, zero and negative branches. The Frequencies of the lines of the Kth single band on the negative side of v. are given by:
- branch: J,K ll _- “J-l,K-1 = “o 2nA J + S(K-1)
I , J = K, K+l . . . . (2.20)
J,K 0 branch: uJ K 1 . - = v. - $ ER-4) (2.21)
- 25 -
+ branch: J-l K -II
‘J, K-l = ‘o --
2oA -J + (K-j)
1 ,J = K+l, %+2, . . . (2.22)
hate that the allowed values of J start with 1: and K + 1, not zero, fol- lowing the structure rule .laK. The intensities are riven by expressions similar to those for a parallel band.
The overall picture of the band for a value of (A/C) = 5 and o = 0.018 for the vlbrational transition 00’0 to 01~0 can then be built up as shown in Fis. 1 (d). Single bands are shown for values of K up
to three.
In the special case of the moment of inertia C approaching zero the central zero bands of all the single bands move to the frequency v.
and so the resulting picture is as in Fip. l(e)., with a very stronp central part of the band.
For a perfectly harmonically desijined molecule the actual band would consist of several of these bands (shown in Fig. l(e).) exactly superimposed, there bein? more bands at higher temperatures. At the same frequency w would bserve 0110 - oooo, 3 2
for example a sunerposition of the bands 05 0 ++ 04 0 , 3331 - 3221. . . . .
However, since in the real molecule the potential wells are of the Morse type the energy levels come closer toEether as the molecule moves towards dissociation transitlon 0550 + 04 2
so that the freouency of the band emitted or the 0, say, is smaller than that emitted for 01 f 0 + OOOO.
Thus in observing the emission of radiation with increasing temperature the vibration bands are seen to spread into regions of longer wavelenrth. To complicate the harmonic spectra still further the line spacing of the positive branch decreases and that of the negative branch increases as the quantum number increases, due to centrifural stretchins, (Zl), and Doppler broadening causes the spectral lines to spread into a continuous SPectrum.
It is worthwhile to note that for CO2 the frequency of the n2
(bending,) mode is very nearly half the frequency of the nl(symmetric
stretchine) mode. This leads to resonance and a zood approximation can be made in calculating the intensities (see the method of Section 3(b) for intensity calculations in the 4.3~ region).
The effect of the shape of the blackbody radiation spectrum on the relative brip,htness of the vibration bands is shown diagrammatically in Fig. 2. Fip. 3 shows some absorption bands at room temperature, all of which are listed in Table 2 top,ether with other important bands.
- 26 -
3(a) The Result of the Application of the Line Summation Method to the
Calculation of the Total Relative Band Intensity of the Carbon
Dioxide 4.3~ Band
Referring back to the expression for the line intensity for a diatomic molecule, we can calculate the equivalent expressions for the parallel band of carbon dioxide.
I p. : The line intensity Riven by the transition (n, nt n3J+n; n2 n; J’) is
P S(n1n2n3. J + n’n”’
6n 3NT”
1 n!,J’) = , 3kc2Q;O; 1 Rn 123 n’n J + n’n”’ 1 2 n;.J’(*
W’ + W’ a” R hv
xe kT (1 - e -izj, (3.1)
where JP the frequency associated with the indicated chanse in quantum numbers,
I 123 Rn n&n J + nio;“n;J’12 = the corresponding matrix element.
NT = the total number of molecules per unit volume per unit pressure,
= the degeneracy of the upper state, with rotational quantum number J:
Qr; = the complete rotational partition function, with energy levels WA
.
= the complete vibrational partition function, with enerRy levels W,),
= the amplitude (5) factors corresponding to the rotational change of state J e+J’e’; typically
Q ::+1 = (J 2 fa)(J T !Z + 1)
for e f 0. 4J(J + 1)
- 27 -
To complete the calculation of the total band absorption we sum the above expression over all values of JJ’.roverned by the selection rules b.J = 0, + 1. (11):’ Physically-this renresents addinp all the intensities of-the lines in the band. The result is
? ~ .
9. e’ -- z U(“ln2 “3 * “in; ‘“;) = a. 0’ .J.J ’ S(“l”~“3J-t”~“~“~~T’)
an3fiT W" 1,; =-
3hc2QV 2 ‘; et6 e
-&l-e-E) ( (3.2)
where z = 9. 1 for e = 0, f
2 for c # 0,
J = the frequency of the absorption band centre,
62 = I e R”l”2”3; “i”; e' “;I2 * and
Qv and ‘c! v are given by
w;(n1n2n3) + w; (Pz, J) = ‘2” (n1n2n3U + biR(J),
Q;1(“1”2”3) Q;% J) = OV(nl”2n3~) I’,( J),
so that the sumnation “ver JJ’ becomes easier. The expressions for Ill;,
QG. etc., will be piven in the discussion of Malknus’s paper.
If At = 0, the above expression for the band nbsorptlon is correct, but if Al..= + 1, there are two possible final states to which the molecule can ho and so the intensity of each band in this case will be half that for At = 0, but there will be twice as many lines since there are two separate transitidns possrble.
1.3 (b) A Special Method of Calculating the Partition Function of
Carbon Dioxide in Application to the 4.3~ Infrared Rand (13)
It will now be shown that the band absorptions for the various bands in the 4.3~ region can be related, using the harmonic approximation and an assumption concerning the frequencies of the fundamentals. The final formulae will be of use in uredictinp, the total band strength of the 4.3~ bandsand will also Rive us information as to which transitions are most important.
- 28 -
(9 Calculation of the Partition Function
The asymmetric stretching. mode in carbon dioxide has its levels of excitation classified by a change in the quantum number “3. men
the other two quantum numbers remain unchanged and n3 changes by unity,
emitting or absorhint a photon of radiation, the transition of states p,ives rise to the 4.311 band.
The usual expression (IO) for the vibrational energy levels of a linear triatomic molecule, taking into account the anharmonicity present, is
h(nln2n3z0 = hc i
wl(nl + A) + w2(n7 + 1) + W3b3 + !)
+ Xllhl + f)? + “22(n2 + 1j2 + X33(n3 + 02 + 8229.2
+ y2(n1 + I)(n, + 1) + x23 (n2 + l)(n3 + 4, + X13(“l + ?)(n, + 4) + . . . I
(3.3)
where w 1’ w2’ w3 are the wavenumbers of the fundamentals and the x.. ‘1
are anharmonic correction terms Fiven by
Y1 = -0.3, -1 x22 = -1.3, x33 =-12.5, (cm. )
x1, = 5.7, x23 = -11.0, Xl3 - -21.9, (Clll. -5 L
and z22 = 1.31 cm.-l.
The fundamental wavenumbers for this empirical formula are
Y = 1351.2, u2 = 672.2, w3 = 2396.4 cm.-l It follows that for a
transition in which An3 = 1, the wavenumber involved is
&1n2n31.m1 “2 (n3 + l)l) = $n1n2(n3 + 111) - E(*ln2n31); $
=&I 3 + 4x13 + x23 + 2x33 + x13nl + X23n2 + 2x33n3 + . . . . (3.4)
Now the fundamental wavenumber wl is alnost twice the fundamental wave-
numhcr w2, and so the states which have the same values of (201 + n,)
and n3 have very nearly the same energy (13). The states with the sam,
- 29 -
value of 0 still perturb one another, and each pair of states would have to he studieu separatelv, leading to a mammoth task. The alternative is to zrow the two ouantum numbers n and n into a new wanturn nunber ,_ 1 2
I
n = 2n1 + n2 ,
and to define a wavenunber z such that
0 = l(luJ, + u,).
We also note that 2x to a ,-ood approximation, so that if we define 23 = 73
we have
and
YJ = IGX,, + xz3)
;;n = x 13”l + X23”2
wn =wlnl + w2n2.
The transition wavenumber then becomes
w ““3
= (nn3 + nn3 +1)
= w3 + jX13 + Xl3 + x 23 + 7x33 + xn + 2x33”3 (3.5)
Also the energy of any level relative to the ground state is in the harmonic approximation:
. . ’
E (II, n2 n3 e) = “lnl + w2n2 + w3n3. s 9 .
This becomes E(n, n,) = ‘;n + w3n3.
The vibrational partition function is given-,by_
.“2
Q”(T) - Y P 7 z _ E(“l”2”3 ) n =O n -0
=i e kT 1 2 , (3.6)
where the degeneracy gt, dependent a” e, has the values
- 30 -
We can separate out the partition function due to asymmetric stretching, this is
n3w3hc
Q3(T)= “‘, e- kT
w hc 3 = Cl- e - -F-)-l
3 (3.7)
The remaininS partition function is
Y 7 F’ (Wl”l+w2”2)hc
Qo(T) = nl=O n -0 2
tF{ 0 v- kT 1 . (3.8)
\!e can now change the order of sumnation, we sum over all values of n
so that after sunrming over e,
1 and n 2 such that 2nl + ” 2 = n, and then SW
over all values of n.
** Let I: f(nln2e) denote the sumration over e (even or odd)
followed by the sumnation over all values of n1 and n2 such that 2nl+n2 = n
(II even or odd), i.e.
“* =;
x f(“1”2a) e”en ” -o 2
“2 n even 1 (n-n,), “2d .,,; epo f
1 (3.9)
n = z ;’ odd n -1
2 odd e-1 1 h-l,-“,), n2e
1 n odd
Then, since wl”l + w2”2 = wn, the partition function is given by
Qo(T) = ,,g t* ‘;nhc
gp- -ET
(3.10)
- 31 -
The two summations are ouite straifhtformward and are accomplished by treatinc P and n odd and even separately. The result is (13)
;hc :hc Co(T) = (1 - e - E-)-3 (1 + e-,-l
27&c ;hc -- = (1 - e IrT )-’ (1 - e - kT) - 2 (3.11)
which can be see” to be the harmonic oscillator approximation partition function obtained by replacin:: w1 by 25 and w2 by iZ. Obviously we could
have put ALI = w = T; straight into the harmonic approximation to derive this result: but2 since the bands for whrch n is constant fall very close to each other in the spectrum, we can predict these individual band intensities much more easily using the above formulation.
In the 4.3~ band An1 9 An2 = 0, and so AL = 0 also. So the integrated absorption coefficient is given by (3.2), that is:
8n 3 U(“l”2”3e~i”;“;~~ = -
E(nln2n3t)
3hc NTB2w’gee - kT
W’hC -kT , 1
wherew’ is the wavenumber of the band centre. For a transition which iqvolves only the quantum number change (A” ( = 1, we can show that 6 a(n3 +l) in the harmonic approximation (as 3. I” the case of a diatomic
molecule in Section 2 (a)).
We can ‘row sum the integrated absorption coefficient over all values of k?, n
1 and n 2’ such that 2n + n = n, and we shall derive
1 2 the integrated absorption coefficient for the group of bands for which n is a constant and n 3 changes by unity, that is
a nn CT) - :* u(nin2n3e + n1n2n3 + U). (3.12) 3
In order to determine the relative intensities of these bands we need also the fractional population N”“3(T) of the level for which 2”l +n2 = n and n 3 are fixed. This is
nhc; n --
N”“3(T) P 8 e kT e
Q"(T) (3.13)
- 32 -
where g" is the de eneracy of the state for which n=co"stant, and en "*
-z% = f t ("+a 5 n even, : (n+l) (n+3) n odd. 1
(3.14)
Thus,
nGhc n3w3hc --
Nnn3(T) = p," e kT e - kT
w hc
I :hc 3 x l+e -FT 1-e--F
3 . (3.15)
The fractional population of some of the lower vibratronal states defined by the 4 internal quanturr numbers nl,n2,n3,k?, is show" in Fie. 4. (14)
It follows that we can estimate the relative intensities of the band groups:
ano "whc aoo(~) - (; * q,)e kT
nwhc -- - - = g"e kT = N"'(T)
N"(T) ,
and
a”“3(T) n3w3hc .= (n + 1)e -
a,oo3 kT - (n,+l) 3 ,
‘nn3 (‘0 so that- = (n, + 1) ~~~~~f) = (n,
(6 + n3w3)hc + 1)g" e- kT (3.16)
For a perfect gas at ordinary pressures; NTal/T and
w hc 3 a",(T) oc 4 N"'(T) (1 - e- - kT ) .
Therefore, o hc 3
annl (T) TO --
=A. Nnn3(T) . 1 - e kT ,
'nn3(To) T N*"3(T,) w hc 3 -- 1 -e kTo
(3.17)
- 33 -
for a given reference temperature To. Thus, we can find U""3(T)
ann3(T,)
for all n, “3 and hence the overall ahsolute band strenrth if we
know aoo(To). At To the intensity we observe is
T Y a -- "00 "2-0
""3(T,) = aoo(To) T Y a""3(T") n-0 " =o
3 aoo(To)
“:hc n3w3hc
= aoo(To) ( ‘t n-0
g”e- kT,)( ? (n, + l)e - LTo ) n =o 3
= aoo(To) Co(To) ? (n n3w3hc
n=O 3 +l)e- l.T
3 Cl *
The summation over n3 can be shown to be Five” by
n3w3hc w hc 3 w3hc w,hc
? (n3+l)e - kTo - -y)e --
ln(l-e -L
n3-0 - Q3(To) 1 - (1 - e kTo 1 LT,,) ,
which can be calculated. The integrated band intensity of absorptioh at temperature To will be give” by
w3hc w3hc -- --
T n =o n i. %q (T ) - aoo(To)Qv(To) 0
kTo)e kTo
3 w3hc
-- ln(1 - e kTo ) 1 (3.16)
Thus one measurement of absorption at room temperature enables us to calculate aoo(To), from which we can find the integrated band intensity
at all temperatures, using the formulae of this section.
The work of Penner (1). after derivation of (3.7) consists of comparing the ratios of 6’ and of a(n1n2’n3 + “i”le “5)
for the g2-fundamental and the combination transition
- 34 -
(* t+1
1’ *g n3 * (n, + I), (n2 + 1) 9 “3) *
1.4(a) Radiation Laws and Definitions
In (1.6) the relationshlp between hemspherical spectral emissivity, radiation densitv and the spectral radiancy for the surface of an opaque solid was plven. We mav define the hemispherical spectral emissivity of an opaque solid surface as the ratio of the spectral radiance which it emits into solid anele 2n to the spectral radiancy of a blackbody at the same temperature,
i.e.
R ('0 E w
CL, R:(T)
(4.1)
where Ro(T) is the Planck blackbody radiation function (15) based on wave nur.ber.w It can be shown that the absorbed spectral radiant eneqy ner unit area 1s eaual to the emitted spectral radiant enemy per unit area if the surface is in thermal equilibrium with the radiation. This amounts to saying
~~(‘0 = c,(T) (Eirchhoff’s Law (16)), (4.T)
where crui(T) is the spectral absorptivity.
Kirchhoff’s Law in this form applies also to non-opaque substances (l), except that both au and E w must be evaluated for the same thickness. It
follows that if P, is the spectral attenuation per unit area in the radiation per unit apical path lenoth, the emissivity of optical path length dX is siven by
~~0) = Pu(T)dX, where X=pl (atm.cr.)
or Rw = R”P dX. ww (4.3)
i.e. The spectral radiancy of any substance eouals the product of the spectral absorptivity and the spectral radiancy of a blackbody.
Consider now the optical system of a -slab” of yas with optical thickness Xo = pl, as in Fiz. 5. The change in transmitted radiation
occurring in a depth dX is due to:
- 35 -
(i) the emitted spectral rndiz.ncy in d:\, Pw(Tp)dXR~(T,) ;
(ii) the attenuation due to absorbers in dX, Pw(T,)dXRw(X). r
The net increase in radiant power noviny: from left to ripht in Fir. 5 1s therefore
dRw(X) = Rw(S + dX) - Rw(X)
=
Leadins to the result that
Rut&) = R;(TS) (1 - a - ‘uJ(~:)~~) + Rw(0)e - PW(Tm)X 0 (4.4)
where R,(O) is the incident radiation from a light source, WY.
It follows directly from this aquation that if there is no chance in spectral radiancy through the was i.e. Rw(X,) = SW(O) we have
Rw(0) = Rw(Xo) = R;(Q).
Since R:(T) is a single-valued function it follows that the (monochromatic) temperature of the radiation is eoual to the temperature of the ras. This situation is used to determine temperature in line-reversal technique, about which more will be said later.
If there is no incident radiation, i.e. RU(0) = 0, The resultin< expression defines the spectral emissivity of a volume of pas
EIJJ =1-e - PJO
The total energy radiated hy the :,as at a temperature T is riven by
where d is Stefan’s constant and c is the total hemispherical evissivitv.
The spectral radiancy of a black body is a single-valued function of temperaturc. Since in practice there are no black bodies It is useful to define the brightness temperature of a body, which is a function of the wave number and of the geometric position with respect to the source and surrounding media. Between a source and a point of observation there is usually a region of absorbinpc medium which causes spectral attenuation PwXl
- 36 -
If the source has a hemispherical spectral emissivity E,(T) and temperature
T o, the observed spectral radiancy is
R;(To) . cw(To)e - Pw(Tp)X dR
5 dug
where TE is the temperature of the absorbing material and dR is the solid angle subtended at the source by a detector at the point of observation; d, is the wavenumber interval observed at wave number W. If the source
was a perfect black-body and there was no absorption of radiation between the source and detector, the same spectral radiancy would be observed at the point of interest, for a body temperature Tb, where
R~(To)~w(To)e - Pw(Te)X $j dw = R”(T ) g dw wb2n ’ (4.6)
Tb is called the brightness temperature of the body. Since cu and
e-w are less than one it follows that To>T b . The parameter Pu(T) is
called the spectral absorption and is related to the integrated absorption S(T) by
; Pu(T) d, = S(T).
The absorption and emissivity of an isolated spectral line will now be studied and the usefulness of the various terms will become apparent. The procedure will be generalised to the case of many spectral lines in order that the properties of bands may be understood.
Spectral lines and line broadening (11).
Consider an atom or molecule in an excited state in translational equilibrium in a region containing, similar particles. This individual particle can change its state, and internal energy, by several mechanisms.
(i) Absorption or emission of a photon without being; simultaneously in contact with another particle
(ii) Absorption or emission of a photon during a collision with another particle
(iii) An upward change of state the energy for which is derived from the large kinetic energy of a collision with another molecule (inelastic collision).
(iv) A downward change of state during a collision accompanied by a corresponding increase in the kinetic energy of two particles (super-elastic collision).
- 37 -
(i) and (ii) are basically radiative transfer mechanisms while (iii) rind (iv) are unaccompanied by any radiative features.
/ l!hich of the above mcchonisms dominates denends on the state of
the $as and the particular way the total enercv is distributed between the various decrees of freedom. For example, (iii) dominates in the relaxation region of a shock wave while (ii), in the emission case, dominates when equilibrium has been reached behind a shock wave at a moderately high pressure.
Since we are interested in radiation properties of Eases. (i) and (il) will be of most importance here, but (iii) and (iv) can be investinated since the redistribution of states they cause influences the state of the Eas and the mechanisms (i) and (ii).
Mechanisms (i) and (ii) are accompanied by observable radiation and for a eiven transition the photon of energy hv, involved can be detected. If we have a very larse number of excited and de-excited particles in translational equilibrium with each other and with an enern,y source, a larcre number of transitions will take nlace. If only one transition is involved the energies of all the uhotons emitted and observed will be similar but not all exactly equal to hv, since there are in the main, three important phenomena which cause an energy spread among; the quanta. These are
(4 Natuial line broadening
(b) Collision, or pressure broadenins. and
Cc) Doppler line broadeninK. .
(4 Natural line broadening is closely related to Heisenberg’s uncertainty principle (2,ll) which states that the product of the uncertaintv of an ener~v state and the uncertainty in the time a particle ii in that sGe has a minimum value given by
Al? At u
If we consider a molecule
probability per unit time
in an upper energy state Eu with a constant
of decayiw, (as inferred by (1.30). the uncertainty in the tine that a molecule is in state E, is the inean lifetime ‘I” of that state. Thus the energy spread of the state is
Since we consider particles in equilibrium with radiation, particles in the lower state will be excited after a mean lifetime ‘p in the lower
state (on the averaRe). The net effect is a snreadinp of the spectral line into a curve of well-known intensity distribution.
- 38 -
(b) Collision broadenin?. As the pressure of a qas increases so does the molecular collision rate which is accompanied by a corresponding increase in the number of collision-induced radiative transitions. This causes a reduction in the mean lifetime of all states causinp further uncertainty m the enemy of each level, broadeninc! the spectral line still further.
(cl Doppler broadeninK. Consider a particle vovinr! with a velocity V, emittinE a photon in the dircctlon of its motion. The observed freauency of the photon (9) ~111 be r.reater than that observed (v,) if the oarticle
were at rest by an amount v - v. = vo: , where 'c' is the speed of lipht.
Since in a gas there is a Eiaxwellian velocity distribution we shall observe a Maxwellian frequency distribution about a central mean.
These are the commonest phenomena which cause spectral line broadenin? in molecular spectra. The shape of the spectral lines in each case, bein? identical for types (a) and (b) is well-known. In peneral we can consider that (b) is pressure dependent and (c) is temperature dependent.
For a single spectral line with one type of broadening dominatinp, the function P,(T) has a definite form though its magnitudes vary accordinp to the total strength of each spectral line. It follows that P, is not the nest useful fom for expressin? the soectral absorption.
Since the line sha?es are well-known: we define the normalized line shape parar;eter b(w) which :ives a wasure of the relative intensity of each part of any 11ne broadened by one of the usual phenownn so that
z b(u) dw = 1 ,
The spectral absorption of a given line (under given broadcninn cowitions) IS then piven by
P” = s b(w) ,
and the emissivlty 1s
EW = l-e - SX,b(w) .
(9 For natural and collision broadening. (also known as Lorentz or contour broadenin?)
b(w) = 2 (IA - wo) 2 + 2 -l ' I (4.7)
- 39 -
where u is the wv~“uml~rr of LIW 11°C centre 2nd r( 0 I.
= nL(n, T) is
the Lorentz hnlf-vidth 7t I:nIf-trelyht 0C the line.
(2) For Doppler Rrondeninf:
(w - tio!71”?
b(u) = c ;, e - a,7
where a D
= aD(T) is the Doppler half-wdth at half-hei-lit.
The shape of the two lines is drawn i; Fio,. 6 for the intensity and half-wiith and it can be see” that (b) tends to
(4 .H)
same zero
pore rapidly that (a), and has a greater ratio of (heipht/wioth). Sowtimes (b) can be qproxiratcd by the square line shane.
Consider now the total wission due to cl sinrle spectral line. This will be Five” bv
7 R,,,‘Sdw = 7 R,‘(l - e - SXb!w))dw
0 0
If the emissivity were unity the total radiation would be
(4.9)
7 Rue dw = “T4 (Stefan’s Law), 0
so that the total hemispherical emissivitv of B line would be oiven by
1 ‘line = 2
$ R,,O ( 1 - e - Sxb(w)) dw (4.10)
Since R, ’ is a slowly varyxnr function it is practically constant across the spectral 1x1~ at wo, so that
Eline =!Liz
oT4 $(1-e - SXb(w))du (4.11)
The intcgrand is close to zero except for a wry small repion Awe close
to the wavenurber wo and the limits of the Integral can be chanred
without appreciable error so that
co ‘line a - I (l-e
oT4 A.wo
- SwJ))& (4.12)
- 40 -
For a single line we define (17)
k’sL = xwp - e - SWd) dw (4.13)
as the equivalent line width. This concept 1s a very useful one and it can be shown that band emissivities are functions of iJsa. and snectral line separation.
In a sinilar wav we can show that for a band, the total (ew,ineerin:) emissivity 1s Fiven by (1)
a =R, ‘band oT4 I
Aw band (l-e -pwx) dw
The values of lIqQ, etc. are well-known for the tvpes of line
given by (4.7) and (4.8) and also for a mixture of the two. However, the calculation of the spectral band emissivity is a mammoth task and the varied assumptions that can be made manifest themselves in the results.
Provided there are a larpe number of lines in the band we do not concern ourselves with the high resolution calculation of spectral enissivity. The individual line shape of each spectral line is exoected to be the same.
From the work of Plass (17) the two basic (and combination) models of a spectral band, and their associated approximations, will be Dresented briefly.
4 (b) The Equivalent Widths of Spectral Lines for Various Line
Shape Parameters (17)
Directly from the definition of equivalent width we can substitute the normalized line shape parameters b(w) and evaluate the integral. Mathematically we extend the limits of integration to infinity.
1. For a Lorentz broadened line we find
%. = 2ly f(x)
SX where x = -
21TUL , f(x) = x eqx IO(x) + 11(x) 1 is the Ladenburg-Reiche
function, (18,26) and IO(x), I,(x) are Eessel functiOnS of imaginary
- 41 -
arPllment. Since xa1 aL
for fixed SX, x is a measure of the spread of the
1 ine. There are two well-l-nown aprroxinations,
(a) For small x the above expression becomes
%, = 2ncttx = sx (4.15)
where xtO.OZq for the approximation to be valid to q per cent.
This is called the weak line (or linear) approximation (1C.L.A.).
b) For larpe x, we find
%k = 2a L(2nx)i (4.16)
where x%12.5/~ to 4 per cent.
This is the strong: line (or square root) approximation (S.L.A.), in which the spectralrline is concentrated very close to the transition wavenumber b:
0.
The Ladenburg-Reiche (3) plot and the two anproximations arc shown in Fig. 7.
2. For the Doppler line shape, the lnteeral can only be evaluated by first makinK the approximations. The results are then
(a) Weak line approximation
%. Y C-1)” ‘b”
c ” n-0 ‘(n+l)l(n+l)i (4.17)
where xD -F z , and aD is the Donpler half-width.
This series 1s convergent for xD<l and also valid to q ner cent when only
the first term is used if xD~O.030, in which case
(b)
“SP. = sx.
Str0n.q line approximation
‘Se = 2aD&i [ ’ + $‘,...] (4.18)
which varies very slowly with xD, c being a constant of order 1.
3. Since the square line-shape is often used as an anproximntian to
simplify the analvsis, width.
In this case
b(w) =
i
l/6
0
- 42 -
it is interesting to calculate its eouivalent
for (w o - la)< w ‘(W. + ta,,
for (Ulo - 16)>w; w>(wo + is,.
XS (1 - e - r)du
xs =6(1-e-T)
= SX, for small xS(=F)
It can be seen that the weak-line approximation is independent of line shape. For large values of x and mixtures of Lorentz and Doppler broadeninr we find that a good approximation is (19)
where a
From this it is seen that even if czD is considerably lnrwr than
aL’ the square root approximation is still valid provided x is larre enouo,h. The nhyslcal reason for this is that at large x the emisslvity at the line centre becomes unity, so that the variations in absorption come mainly from the wings of the line, which are much more effective in the case of a Lorentz broadened line. (See FiK. 6).
The physical meaning, of equivalent width is piven by
which is the width of a line of unit spectral emissivity with the same total emissivity as the original line.
- 43 -
4 Cc) !+odcls of Qx~l Spectra and Calculation of Snectral
I:vissivitv in terms nf Couivalcnt lL!idths of Sinolc Llncs
I:‘e have seen that the total emissivity of a line is rrlated to the equivalent width of-the line. That is
Rzo 0) %(T) = “T4 lQse(T) (4 .?O)
We can also cnl&late 1’ se(T) for an! line qivcn S(T) and h(w,T).
A vatural extension of these arguments includes emission 2nd absorption of larw numbers of lines in a haul. For a hand made UP of snnerrosed lines we can calculate the total fractional ahsorntion h(T) ’ which is the tctal hexvispherical enissivity of the band, m&r the conditions of Kirchhoff’s Law, E hand(T) = A(T).
There are two basic models of an absorption band:
(1) The Elsasser Model. and
(2) The Statistical Model.
(1) The Elsnsser Elodel (20, 19)
The Elsasser hand contains an infinite number of enually spaced lines with eaual intensity and identical line shape parameters.
The fractional absorption of the band is found to be represented in the form of an inteRra which cannot he evaluated in terms of elementary functions, but can be approximated by:
(a) A = erf(iB’x); Valid if ~1.25, 8~0.3 (to 10 per cent),
(h) A = 1 - e -6, ; valid if 8>3, for all x, (to 10 per cent),
where 6 2na SX sx Pd. x =%a =3, d being the line sepa?ation.
If we consider the physical internretation of IJsL viven in (4.19)
it is very easy to see that the fractional absorption for a Lorentz-line band is Eiven by
A= ‘h. e-
= 6 f(x).
This expression is correct to 10 percent if x<0.06Bq2 and
EN.3 or x<O.24-’ and DO.3.
Note that (4.20) no longer applies since we assumed that Rio was constant over a spectral line. llere we have many equally spaced lines and Rz” is certainly not constant across them all.
- 44 -
(2) The Statistical F!odel (17)
Vhereas the Elsasser Kodel is one of a perfectlv regular frequency spectrw, the snectral distribution of lines in the statistical model is random, as is the intensity of each line. In fact position and intensity are p;overned by probabilitv distribution functions. No correlation between lines is assumed, and it can be shown that the absorption is a function only of the eouivalent width of a sinple line of the band.
Define 1I(wl, . . . . . ‘J’J dw 1 . . . . . dw n as the probability that the
first line will be in the wavenumber ranpe dwl at w1 while the second
line is in the ranr,e dw2 at ~2, and so on up to line n. Take the
wavenumber origin at the centre of a band of width D. Define also P($)dSi as the probability that the intensity of the ith line is in the range dSi
at Si. The spectral emissivity of the ith line with a shape b (wi) is
therefore (1 - emsix b(w’) 1 ) which is also the fractional spectral absorption for a” optical path length X. The total absorption A for a statistical band is then
A=l- N(w
l . ..w”)dw 1 . . . dun 7 . . . 7 ‘! P(S;)e -XSib(wi)
0 0 dSi
@ 3” -iD . .._ 1D 7 N(wl...w”)dwl . . . dun ” . . . ” L ~ U P(Si)dSi
(4.21)
Since no correlation between the frequencies wl, . . . . (L,” is assumed, N
must be a constant, and if we normalise P(Si) so that
? P(Si)dS. = 1, 1 0
then since the intensities are unrelated the integral over each line is equal to the integral over any other line, and A reduces to
AD l- {+ I dw 7 P(S)e-sXb(w)dS 1”
-1D o
- 45 -
1;~ further mathematical rearrnnyenent we find that
1D A=l- 1-i 1
" do, 7 P(S) (1 - e -sww))ds
-10 0
-l- [l -4 z P(S)dS -1; (1 - e-S'%(w)) d,,,i" 2
=l- i
1-i Igp ,,(.%a){ n
(4.22)
where "Se D&x) is the mean value of 'J se ,(W over the distribution P(S) 9 .
in the bandwidth D. ? is some standard intensity within P(S).
Now D = nd, where d is the avcrare line spacinv in the band, so that
If the nunher of lines n, approaches infinity while d remains constant, that is D-m, then
l!st cs,a,
A=l-e- d , (4.23)
where lss("',a~ = "i (l-e - sxb(w))du, cl
and v&Co = 7 P(S) VJsQ(S,a)dS. (4.24) 0
These results demonstrate that for any value of n, the fractional
- 46 -
absorption of a band, includinr all overlappinr effects, can be r rcprescnted as a function o. the equivalent width of the spectral line.
It can be shown that provided the number of lines in the band exceeds ten, the predicted ahsorption curves (a nlot of A vs. x for fixodE, where x and 6 refer to n sinfile line in the band) are almost identical (see FiE. 0) for all values of absorption of practical interest. (O.Ol<A<l).
Tt can also be shown that the fractional absorption is nearly independent of the function P(S).
For x square line,
(i) For P(S) = 6(S - 7) , i.e. equally intense lines.
XT - I’S1 = 6(1- e--Q
(ii) For P(S) = 1 e - . e. an exponential intensity distribution. -s
These two expressions for% are very similar, especially if F cc 1,
which is the W.L.A.. In fact, providing we correlate all the values of 3 for the various intensity probability functions, we can get nearly complete agreement for any reasonable intensity probability function P(S,S). (17)
Note that in the W.L.A.:-
m
k’Sp’a) = I SX P(S)dS = XT . 0
A comoarison between models (I) and (2) is shown in the plot of fractional absorption curves zn Fig. 9.
Since, on the whole, molecular spectra are neither a completely random nor perfectly renular set of lines, the obvious eeneralization of the above two models is the Random Elsasser Model (17). This is a random superposition of pure Elsasser bands of varying total intensity and line spacing. The result of an argument similar to that given for pure statistical model leads us to
- 47 -
-1 lA1 1 A,: m A-l-
m N dw l..... duN I . . . I !I P (S.)e
-SiWi(m,)
0 * i-1 E l dSi,
(4.25)
where Ai denotes the line separation of the ith Clsasser hand of which
there are N; bi(wi) being the line shape factor summed over all the lines
of the ith band ; PF(Si)d Si is the probability that the ith hlsasser
band has total intensity in the range dSi at S.. 1
The expression reduces to
s :I A=l- II
i-l (l-+).
i
m
Fere, W z‘,i = I ‘iE i(xi, Bi) P(Si)dSi,
0 ’
(4.25)
with 8. = 2nai 1 T i
ana x. Six 1 =2na i '
m
i7- E,i E o ’ ‘i %,i hi,Bi) P(Si)dSi,
where %,i(~i,Ri) is the ahsorption due to the ith band.
This formulation of the Random klsasser band allows us to find an infinite set of ahsorption curves (i.e. A vs.x for fixed e), between those Five” by the statistical Fodel and the Elsasscr Vodel, denendiw on the mlmber and intensity of Elsasser bands we consider, (see Fig. IO).
Further, as N + - and the average line spacinp d, oiven hy
is held constant, the ahsorption curve annroacheq that of the statistlcal model. For exarmle if Si , ui and Ai are independent of i,
- 4s -
WY :,i
(Xi, 6.) = 1 qmn
$1 so that A = 1 - (1 - z;i)’
- 1;
E s1-e-r
as:4-+- ;
which is a statistical node1 result.
In most molecular hands at high temperatures this number of supernosed bands is so large that the statistical model has been used in emissivity calculations, (particularlv for carbon dioxide).
5. Apnlication of the Statistical ?lodel to the Vibration-Rotation
Bands of Diatomic and Linear Triatomic Molecules, taking the
4.3 micron Carbon Dioxide Band as an Example in Calculating
the Spectral Emissivity.
We have seen in Section 2 that the relative intensity of sinple spectral lines in a vibration-rotation band can be calculated as a function of the rotational quantum number. In Section 3 (b) we saw that the integrated band intensity can be found at any temperature if its value at one temperature is known.
This information allows us to calculate the spectral enissivity of a vibration-rotation band but would involve consideration of each individual line of the band and there may be well over 100,000 of these at high temperatures. There is however a ‘reverse’ method formulated by Malkmus and Thomson (21) for diatomic molecules and adapted by Malkmus (13) for the 4.3 micron band of carbon dioxide.
This method assumes a diatonic molecular model consisting of an anharmonic vibrator and rotator with the first order vibration-rotation interaction.
The energy states of this model can be expressed in terms of vibrational and rotational quantum numbers. The transition wavenumber from state (n,j) to state ((n+l), j’) is given by
w = w(n) + B e j’(j’+l) - j(j+l) ] - ae [(I-I + 3/2)(j’+l)j’ - (n+l)j(j+l)],
(5.1)
- 49 -
where w(n) is the central wavenrlmher of the hand and depends only on vibrational quantum numhcr n, and potential well constants. T, e IS the
rigid rotator snectroscopic constant and =4e is the first order vibrntion-
rotation internction constant.
The rotational selection rule informs us that
.1 J = j+1 (with j, j ‘>O),
l&din: us to two evressions for w, one for each branch of the hand. There is no Q-branch for a diatomic molecule, and for a parallel hand of a linear triatonic rolecule the r)-branch is very weak. The P- and R-branches have lines at the following: spectral positions:
w = w(n) + s cl j(j+l) - 2(i+l)(n+
21;;1? - aef i(j+l) 312)
- 2i(n+ 312) I (5.2)
The upper expression correspx~ds to j’ = j + 1 (R-branch) and the lower expression to j’ = j - 1 (P-branch). By suhstitutinu j = m - 1 into the upper expression and j = - TP into the lower one two identical expressions are derived:
w = w(n) + 2n E - a e e C
m(m + 1) + Zm(n + f) 1 (5.3)
This sinole expression describes both the R- and P- branches by means of the appropriate substitution for m. Enuntio” (5.3) is then solved as a quadratic equation in m leadinp to two solutions
“e - ae(n + 1) - (B e - ae(n + 1))2 - ae(w - w(n))
ml = a e
n e
- ae(n l 1) + (Be - ae(” + 1))2 - Ueb - w(n)) and m =
2 a
e
which ?Ialkmus and Thomson associated with the P- and R-branches respectively.
The expression for the integrated line intensity of A single line in each branch of the hand can be found absolutely as a function of the vibrational and rotational quantum numbers of the initial state (n, j) and
- 50 -
spectroscopic constants (31). Ry substitutinn i = j(m) we can find the integrated line intensitv for the band as a function of m. Further by substituting ml = m (w) and m line intensitv as alfunctwn Zf
= m (w) above we can find the inteRrated (n: w). This operation involves a smoothing
out of quantum numbers, but since there are nanv lines this does not cause any trouble if onlv low resolution spectroscopy is required. Application of the Statistical Xodel of a band enables us to complete the low resolution spectral emissivitv of the band.
Thus from S = S (n-w’, i+i’) we can find
S = S(n+(n+l), W)
and hence the spectral emissivity. ror example in the weal. line approximation, we find, fron Equ (4.23), that
E w P 1 - exp (- ,* :
where now we have
(5.4)
IIerc d,(u) is the mean line separation in the band and is given by
d,,(w) - 2 - ae(n+l) ] 2 - aebJ-w(n)) . (5.5)
For computation of cu by this method physical conditions of
pressure, temperature, and optxal path lenqth are chosen, and for a given wavenumber (S(w)/d(u)) can be calculated usinp, the summation.
Malkmus and Thomson (21, 13) associated the two solutions of the quadratic equation (5.3), that is m and m , with Lhe P- and R-branches of the spectral band. In fact the solition rn: describes both the P- and R-branches up to the band head in the R-branch, The solution m deacrihes only that part of the band beyond the band head. In this region th$ rotation-vibration interaction energy in the first-order approximation is larger than the rotational energy,and hither order interaction terms should be considered. Fiq. 33, which shows a plot of equation (5.3). should help to clarifv this point.
- 51 -
It happens that the intensitv of the spectral lines approaches ‘zero rapidly beyond the band head, so that the contribution to the emissivitv of the spectral lines described by m caused by neglecting the solu $.
is very snall. The error in the emissivitv, ion m is very much less than the main errors
of the calculation. These stem froi’the use of the harmonic oscillator approximation in predictino the spectral line intensities and the use of an approximate spectral band model.
Thus, to within the accuracv of the models used in the calculation we can ner.lect the term
s;+l (Id) ( F )m2
in equation (5.4).
\!e can now proceed with a sumarv of the work done bv Ilalkmus (13) on the 4.3 micron band of carbon dioxide.
The similarity of the 4.3 micron band of carbon dioxide to that of a diatonic molecule has been noted in section 2(b). The 4.3 micron band consists of two types of superposed bands.
(i) Those for which &O, when alternate lines (of odd j) are absent. but the remaininr lines have twice the intensity expected from diatomic theory
(ii) Those for which e>O, when the average line intensity is the same as for diatonic molecules, but there are twice as many lines in the band, (see Fip;. 1). A weak central branch is also formed, becoming stronger as P increases, but since the lines near the band centre beconc weaker as t increases, both effects are ignored. Even at 3OOO’K the averap.e value of f. is only about three.
The spectral emissivity is calculated for both the V.L.A. and S.L.A. and also for Lorentz and D8ppler lineshapes.
In the weak line approximation we saw that I’S8 - SX: leadinp.
to Fse = TX, and now with the arguments laid out as ahove T can be
regarded as a function of wavenumber. Thus
Using. the methods of Section 3(b) we can eliminate the ouantum numbers t. n1 and n7 and replace them by a combined quantum number n and a degeneracy factor.
The corrected equation (5.4) becomes
(5.7)
- 52 -
a where s **y =
“,, ?
Ile I~CUJ hew (1 - e
- “CW”“” -1 w
““3 kT -kT)(l-e kT ‘)
x exp - - t
IlCL 7L(L2 - ix-G-) - ";
ka3’T ( I +z). F: 1 I (5.3)
and d (w) = ““3
:!JxT, (5.9)
where L = I1 - e
cL1(“l + !.) - a2(n7 + 1) - “3(“3 + I), (5.10)
and i: = a3bJ - W”” ). 3 (5.11)
w nnd a ““3 ““3
are defined in Equs. (3.5) and (3.16).
The constants Re, al, a,, a3 are riven in earlier papers
(for examle see Ref. 22). Since a1 = -2a2 << Be, P!alkms neqlected
contributions due to al and a 2 . liere we have considered only one root
of the auadratic equation (5.2) when the approximation to the vibration rotation interaction is most valid.
I” the S.L.A. for Lorentz-type lines we saw
w = 3a1(2nx) 1 1 sa 1
= 2(ULSX) ,
and it is necessary that (a) lines crouped toEether are not considered exactly superposed, (b) the 1 = 0 bands have a spacing and intensity different from the other bands. Since p. = aL(p,T) ap, we put aL =a op.
p beinp. the pressure in atmospheres. ’
Therefore cu = 1 - exp(-2a. 1 2 1 &w) (P 1) d(w)) (5.12)
- 53 -
4 m m where yJ = ,fn F (Ir”+
@nn3 (w) +
“;=O d nn3 (w)
where p. " is n,iven by (3.14). For a mixture of oases with a total ~rcssurc TT and CO2 partial pressure of p
P' in the weak line anproxiration (~1)
becomes W) ' and in the stron: line approximation (r'l); beromes (p p TP
1);.
For Dorplrr line shape we had For Dorplrr line shape we had
Assunino the statistical node1 we find
[ -
m m = 1 - exp x X d
n=O "P ““3 1
(5.14)
(5.15)
m
where !! *s9. = s* (-1)" ““3 nn;
x. z [ Sk; Cw) p$ ] " (5.16)
m=o
d* ““3
s;“3(w) = 2”. . s r, ““3 tw) ’
and d* = 2 ""3 n' d (WI, ""3
which were given previously. c
Vallmus calculated the emissivities in both !!.L.A. and S.L.A. for T = 300oR, fiOO@I:, 120001:, 1900°K, 240Oo1., 3ooo°K. For purpose of calculatiq ooo (T) hc assumed.tbat the total band streqth (23) at 300'1:
- 54 -
was 2700 cm. -2 -1 atm. . That is
m m z c a (300") = 2700 cn~.-~ -1 am.
n=O n =o ""3 3
a Since ""'
CT)
a00 (To) is known, " 0 ), and a
"5 o nn3(T) can be found.
The value of ~~(24) was assumed to be 0.064 cm -1 at 3OO'K and to vary as T -1 . The accuracy of this value is questionable as it is based on measurements of the nitrogen-broadened 15~ band. The values of nn
a. considered were limited bv takinp into account only those
states for w lch the fractional population Nnn3(T) was greater than 10 -3 of the maximum value of aJ"n3(T). Since the intensity of each "averape spectral line" was calculated it would have been just as easy to leave out all those lines weaker than a given strength, rather than leaving out allstateswith less than a given fractional nopulation, esnecially as line strength is a function of dipole moment as well as level population. Also the presence of 1.1 per cent of C13O162 was ignored. The error here would be more noticeable at lower temperatures.
Use of the above formulae enables us to convufe the exponents involved in the expressions for the emissivity.
Fips. 11 to 16 show the variation of -5 Id and 2ao@id
with wavenumber for the temperature already specified and also the variations with temperature of 'id for wavenumbers w in the range
1900 f w zj 2390 cm. -1
Fias. 17 to 20 show the emissivity curves for weak line approximation and Doppler line shape for the temperatures 300; 600; 1200; 15OO'X. It is seen that the pure Doppler line shape provides a lower limit to Ed, for any
given conditions. The apreement between N.L.A. and "we Doppler emissivities is seen to improve with i"creasinF: temperarure and decreasing optical path length.
Figs. 21 and 22 compare the W.L.A. with that derived by Plass (27) for T = 1200°K and 2400'K. The discrepancies are alarming, particularly at the higher temperature, when the two curves are sewrated by a factor of about 15 for any riven wavenunber, but the Zenera shape is seen to awze fairly well. However Plans's results have bee" shown to be inconsistent with the harmonic approximation in completing the integrated band intensities, (28). The same tendencies are found by cornpain? the S.L.A. in both Plass's and PIalkmus's work (13).
FiEs. 23 to 25 compare the emissivity with Ferriso's experimental (supersonic burner) results (28). The W.L.A. is see" to improve with temperature. This we expect since the number of lines increases rapidly with temperature. Fig. 23 shows also the S.L.A. computation of Ed, which is
- 55 -
see” also to nrerlict a valur lamer than the observed one. In fact in the absence of Doppler broadcninr, each of these approximations separatelv nrovide an unper limit to cu. IJhen the spectrum is one of a few strong lines sunerrosed on a bacl,xound of many weak lines, neither :!.L.h. of S.L.A. is a qood approximation. I” this case the value of the summand in the two cases is computed separately for each superinroscd band, and the lower of the two values is taken. i.e. we take the srnller of the two term’:
s Pl
““3 (lJl) d
““3(W) , an?
Since the error in both terms is known to be such that both terms are too large we shall eet a better result by always takinr. the smaller one. TLe procedure is “ot purely empirical, but asslsns the most valid a;nroxination to each band and of course rives a more accurate result than either ‘d.L.A. or S.L.A. alone, as can be seen I” Firs. 23, 7.6, 27 and 26, where the shaqe of the curve is reasonably well predicted. The asymetric peak in the experimental curve of Ferris” at 2400°K may be due to over-correction for atmospheric absorption.
Fips. 26 and 27 compare the results with those of Tourin’s yas-cell observations (14) at f2OO”K. The two figures are different onlv in that the carbon dioxide is “ressurized in Fix. 27 by O.E55 atm. of nitrogen. The l:.L.A. yields the same curve for Ed whereas the other anproximations are better I” the case of pure C02. Thus the nitropen does not broaden the individual lines of the band as much as ow models. As the value of do at 300°K is questionable this is one of the possible sources of error.
Fin. 28 shows similar conparisons at a different ontical nath length t4 aurch’s results (29) at 1200°K. and Pi-s. 23 and 30 coware these results in the strong: and weak line npproxil”ations to those of Oppenheim and Cen-Aryah (25)) who use the statistical model to calculate transmittance (Section 6).
It is worth notinp that in the strong: line apnroximntion, where
E p 1 _ e- 2a 1,1(p71)1/d
0 ;
w
the enissivity is not a function of X = pl, so that the elementary law of absorption (Lambert-Seer’s Law) is not obeyed. The Lambert-Beer’s Law expresses tW = cW(pl) in the fom
- 56 -
where k w is the spectral absorption coefficient.
Thus a consequence of the S.L.A. is loss of linearity in optical path lenrth.
It is evident that the mixed weak-stronq line annroximation is more accurate than either W.L.A. or S.L.A. alone, though the estimate of enissivitv is still too larp,e. However Dure Doppler line shape alwavs under-estimates E IJIB So that we can at least calculate the two limits between which the true emissivity curve lies.
6. Direct Applicatwn of the Statistical Node1 at 1200°K, in order
to Verify its Validitv ., and to Derive an Expression for the
Spectral Absorption (25)
Consideration of quantum jumps and individual spectral lines is laid aside and the statistical model, as described in Section 4, is used to attempt to find from soma simple absorption exneriments the empirical laws poverning the band radiation. The study is confined to Lorentz lines only, for which the full Ladenbury-Reiche function is used to replace its aspptotic linits of the Neak and Strone Line Approximations for the equivalent width.
The situation considered is a cell of gas of length 'E' at a constant temperature of 1200%. The experimental variables are the pressure p and 9. Only pressures oreater than 0.071 atms. were used in the exneriments, since below this, at l?OO'K, Donpler broadenin? has a siv,nificant effect.
The statistical nodcl was riven in Section 4, where we had the result that the fractlonal absorption A could be represented by the expression (19:
(6.1)
!.%ere
IiSe CT. a, ", 1) = $ Wse(S, a, pp ) P(S, S) dS,
P(S, ?) being the intensity probability function and? some nean intensity. Since the dependence of A on the actual form of P(S, 2) is small (17), we shall use throup,hout the distribution
P(S,X) = 6(S -3,
where 6 is the Dirac g-function.
Also in (6.1). d is the nean line spacinn in the band. Since there
- 57 -
am many lines in the band, ncross which we consider the Planclr rx'intion function to bc constant. w shall consider that A, " (a. S) and II arc SV
sow, since we arc studying WI-C Lorentz lines for ullicb the half-width is proportional t* pressure. that is
it follows that !? = II * se,w SE,2 * where the superfix 0 denotes all nuantit~es
at unit *ressure. Also, bv definition
s = s;p . w
Assuninr the Dirac intcnsitv probability function, it follows that
(5.2)
SOP, s 9. For mre Lorentz lines 51 '
SP ,w - Zno;f(x(,). where x1,, = -!- 211n"
=w IJ, 2lm,
for this special case of p = 1, and f(x,) is the Ladenburq-Reiche function,
shown in Fir. 7, and Riven by
f(xJ -x
= xue w 1 I"(XJ + I1(xw) .
I
Therefore, for rurposes of calculating Psi w , x , fJJ
is independent of
pressure.
Denotine the spectral transmission by T it follows from (6.2) that
w
-en T, = - 9a (1 - A,)
and a plot of - 9.nTw versus p results in a straight line thr0uo.h the origin.
- 58 -
however this does not constitute a nroof of the statistical model since the simple Lambert-Ceer’s Law also states that
-PAI T w = P kue ,
leadinc to a linear plot.
Comparison of the above tw absorption laws shows that we should also consider k as a variable, since it is implied that
!kG! is a function of k if the laws are to aeree over some range
du of (p,f). In fact this dependence on E is ouite pronounced and is represented in Lhe Ladenbury-Reiche curve.
Thus since
2na0 T
w =exp (4-p ) F f (Xu) 1
w (6.3)
P.nT a plot of - 2 versus i for a fixed frequency should follow the
P Ladenburp-Reiche curve. Once this plot is shown to agree with the theoretical curve we have proved the validitv of the statistical model.
The function WsT w is mathematically determined by the FarilPeterS
S; and a;, ,I
integrated intensity and h$f-width at unit pressure. Thus
A w is fully determined by (2) and cdw -) which may be read off from the w w
Ladenburg-Reiche curve, once constructed.
The following procedure is useful
(9 A recording of the absorption spectrum of the band, measured from a cell of fixed k and arbitrary Pressure, for the pure gas is made.
(ii) Equation (6.2) allows a calculation of G) for many values of w. ( d w
(iii) (i) and (ii) are repeated for a number of cell lengths, and for each wavenumber the values of ws;l,w are plotted against a on a lee-log
d w scale, that is a graph of
2naO 2na0 2
dw f(xJ vs. E(- -g$ ,Q for fixed W.
w
- 59 -
(1-J) Flttlnm these c~lrvcs to that of
(v) These ruantltles. evnerimentally established, cive 115 R value of A L.1 for any value of pressure or Feometric nath lenoth since
(6.4) .” S”
A ‘I = 1 - exy C- 2n$ 7 Eg-c--$, ) 1 w ‘- I.!
x 0” The values of ($) and ($1 can he wed to nredict the effects of
01 -as mixtures, provided that at is corrected to include Foreirn-*as broadenlrp
effects. For examnlc, if we aswme that the half-wiiths ?ue to self NIC’ foreign-pas broadenin? are equal we find
where 7 ‘T 1s the total pressure, and if in a series of experlnents 9
is constant and “I ?T
is cons tat, it f0110w that !J ” s 9. , w is unchnnzed, since
(6.2a)
The gas-cell experiments were perforred by Oppenheim and aen-Aryah for five different cell-len$hs (25). The results are plotted on the Ladenhur*- Reiche curve (18) in Fio. 31, showinf Food avreement at w = 227; cm-l. Thus the statistical model is a pood a-proxination to the 4.3 micron band at 1700°K and the dependence on (II, 9,) is also predicted. This elevated temperature was of course used to increase the nur?ber of 1~~s in the band, makinq the statistical vodel a more feasible approximntion.
In the experiment it was necessarv to record a smoothed absorption curve, since it is the overall band shape that is important, and not local peaks due to a lnrre spectral line. Thus the slitwidth of the spectrometer was increased until a reasonabln smooth curve was observed. This slitwidth was 2 to 3 cm ml and the bandwidth investigated was 2200 tz12400 CT-~. In this semx T w
is an averape over a bandwidth of 2 to 3 cm .
- 60 -
The Full results arc? shown III the form of Franhs in the oripinal p‘qwr. The curves shown in Fix. 3’2 are the resrllts of the computations outlined above where the curves have been replotted for more sutabla path leneths. These curves may be used to predict the absorption in experiments usin? PBS mixtures in the following. way:
(3 For a given path length P,’ select a figure For which 12 $ e’, where e is the path lenpth in the fip;ure.
(iI) Select an experimentally convenient pressure n from the firure.
(iii) For a RBS mixture for which the total pressure pT = n and the partial pressure is p’, such that p’ = p
(j-f) 9 the transnittence is
piven by the curve selected in (i), (ii).
This nrocedure neglects foreirn-pas broadeninp effects.
Eiost workers, prior to Opnenheim and Ben-Arvah, (25) who meas~lred carbon dioxide absorption for the 4.3 rricron band assumed the validity of the Lambert-Beers Law, which predicts a linear dependence of -PnT w on pressure. This dependence was indeed found, but the qradients of the lines predicted very different values o f the absorption coefficient, for various values of nressure and cell-len@h and also various methods of heatins. Thus Tourin (14) (quartz xas-cell) found a value of k = 1.09 cm. -1
-1 w
am. at 4.4 micron, while Steinberg and Davies (30)(ShocL Tube) found
k = 2.9 cm. -1 atm. -1 at the same wavenumber and conditions.
Assuming the statistical ‘model it is possible tq reconcile all these published results, using (6.2) or (6.2~1) as the correct absorption Law. These results are correlated on the Ladenburg-Reiche plot (18) (Fig.31) showing that all the points fall reasonably near to the theoretical curve. This also confirms the method of accounting: for the effects of foreign
L”T gases, which consists of plotting values of - 2 , for a Riven e,
P partial pressure p’ and total pressure pT with an abscissa of @=I).
.Q.“T PT The ordinate of each point was taken equal to -2 .
‘T ‘ ’
Thus,Eiven the values of two of the followinp:
which have been defined in this section, “‘se,w
we can calculate the absorption
A w = 1 - e- dw ,
as a function of temperature and wavenumber.
Of course. knowing the fractional absorption allows us to find the emissivity using Rirchhoff’s Law.
- 61 -
s . Conclusion nn<l .? Grief Discussion oi SyccLral and Vo~ccrllnr ~loclrls
l'c have seen tllnt tiwrc PW two qywrentl- nu1tc d1ffcrent petllods available for calrulatin~ ti-e sncctrnl enissivitv or clbsorntiw ty of a vibration-rotation hand.
The first and rrnst imyortnnt methoJ is the fundnmcntnl one of considerins individual runntum rrcchanical transitions, and perfornin? sumrations over various dercnerate states in each vihratlonal level.
The second mcthofl is that which is nwlicd direct to thr spectrum, innorinm all ouantum mechanical transitions, and considers n r-odcl spectrum - that of the statistical model for an infinitr number 06 lines.
Each method of attack rewires sow fundaFenta1 l~~ovlcdqe, which cnn only he obtained exnerimentally.
In the fundamental pethod this knowlcdre is in the fan- of snectrosconic constants concerninr:
(9 enerpy level maonitudes and separation (?2),
(ii) anharmonic potential terns (6, 7, S),
(ill) the effects of nultinolar transitions (4),
(iv) dipole and multinole xwtrlx elements (4).
Even then the first anproximatlon is Fade in the perturbation theory calculation of Section 1.
For the statistical model TE require more information than this hut it is of a simpler nature and can bc found from some simple fairly low resolution spectroscopic experiwnts of the tvpe mentioned in Section 6. These experiments would be required at different temperatures. Each set of results would be required to wtisfy the statistical model, which can then be used to determine the pamceters given on page 60 as a function of temperature. The model then rives us an errpirlcal formula for the absorption
A = A t, w, :(u, T), ;(w, T) I
.
Of course this is for pure Lorentz lines. However, for shock-tube work of fairlv short geometric oath length, the pressure will need to be large enouy,h to eliminate Doppler broadeninp effects in order to pet a reasonable amount of emission.
If one method is carried right through to predictino A say, as above, then the other nethod can in princinle be fitted to the empirical function A, and the reouired spectroscopic knowledge can be obtained for the fundamental method,
The spectral-model anproach is a very common one, those mentioned in Section 4 (c) beinp a sample of the more sophisticated tvpe.
- 62 -
The full set of models are:-
(i) The Box Approximation or Box-Hodel
This involves the definition of an effective bandwidth Aunn,
together with an nvera~e absorption coefficient xi. The result is that the absorption profile 1s considered to be of constant height, with vertical sides, havin: the same area as the measured absorption. The inteRrated band intensity (see Section 3(b).) lends directly to this model as soon as the band is given a width, which is arbitrarily defined for this model.
(id The “Just-Overlapping” Xodel $
Here the averape absorntion coefficient is approximated by (2) w
and is considered a function of w, where d, is the local line soacinp. The rape of validity is roughly the same as (i) but mav be somewhat superior for moderate nressures and optical path length. The analvsis is more complicated but leads to an unambiguous value of effective bandwidth.
(iii) Non-Overlapping Lines with Lorentz-Tvpe Lines
For this model the analvsis is as accurate as for a single line, but is available only for low teqeratures and high resolution applications and is not applicable to the hiphly populated infrared hands of carbon dioxide.
(iv) Son-Overlapping Lines with Pure Doppler Broadenin?
As for (iii) but at hioh temperatures and low pressures.
I (v) Non-Overlapping Lines with Nixed Lorentz-Doppler Broadening
This leads to a full range of teqxrature and pressure application - but only for hip,h resolution.
(vi) The Elsasser llodel
Equally spaced and equally intense spectral lines with arbifrary overlapping, and either of the two main line shapes.
(vii) The Statistical Model
Lines p,overned by random probability distributions for intensity and position in the -band.
(viii) Random;Elsasser Node1
A combination of (vi) and (vii), such that a complete Elsasser band becomes treated like one line in the pure statistical model.
(ix) Partiallv overlappinp spectral lines with Doppler-broadening.
(4 A locally-applied statistical model for arbitrary over-lapping
- 63 -
and with pressure hroadeninv. This amounts to considerlnr eouivalent width as a function of wavenumher, as in Sections 5 and 6.
It should be pointed out that the fundamental method of attack on the problem is also forced to become a model anproach since we cannot predict the intrn-molecular paramctcrs precisely. In this case the exnression for the band is tbst of a molecular model, for example, the harnwnic oscillator model, rather than a less realistic spectral vodel which does not consider why the lines are there at all.
Spectral models were introduced and investipated mainly as a scane-goat at macrosconic level, because molecular models were too siwle to nredict realistic spectra, or too conalicated to handle: at all.
kwever, the vethod of I4alkmus & Thomson (21) utilizes both a spectral and a molecular model and produces results which agree with experimental data. This is no doubt the best method of solution of the problem so far devised, but ivrovement on its accuracy while keenins the basic method the same would require alsebraic solutions of hipher order equations.
This work is part of a thesis nresented to the University of Nanclrester in support of an application for the derree of M.Sc., in September 1965. I am grateful to Dr. H. K. Zienkiewicz for his active interest and encouragement in the preparation of this work, and to Elr. E. l!ild for his helnful discussions on difficult theoretical points.
I am indebted to the Scientific Research Council for n maintenance prant during the compilation of this worh.
- 64 -
(a) P=O’
d-d I Y
L-IL (b) 4 sqiall
-~ _ _. _ .__- -..
(c) P large
FIR.1. Theoretical spebtral line positions and relative lntenslties
for, typical component bands of the carbon dioxide 4.35 band.
Fig. 164
- 66 -
Fip,. 2 (a)
I I
.m
Fig. 2 (b)
- 67 -
Fi$:. 2 (c)
Positions of centres of vibration-rotation hands observed
at room tellTerature. The desienations vs (very strong), s (stron?).
etc. are those of Herzhere (10) and correspond to those of the first
column in Table 2. Also shown is the radiacy ratio (Ri/Rzmax) for a
blackbody at 30O’K (2 (a)), GOOOK (2(b)). and 15OO’K (2 (c)j.
- h8 -
Table 2.
r U: pper state
01’ 0 02" 0 02'0 10' 0 03' 0 03' 0 12 0 11' 0 oc 0 12'0 20'0 03' 0 1s 0 GO" 1 00" 1 02"l 10' 1 04- 1 1r 1 20" 1 06" 1 14" 1 22" 1 30" 1 OB 2 02": lo" : 00" = 02"; lo" = L
T
,
, , j i , i
nfrared Sands 8orrer state
00’ 0 0s 0 OS 0 OS 0 02= 0 02'0 02'0 02" 0 OS 0 OS 0 OS 0 00" 0 00" 0 oo"0 00" 0 00" 0 00" c 00" c 00" 0 00" c 00" c 00" c 00" c 00" c 00" c 00" c 00" c 00" c CO" c 00" c
of co
r Y,
Y Carbon .LuL.%nt.A huenumb*? (cm-')
667.3* 618.1 668.l.e 72o.w 596.5 646.1 741.7 791.3
1880.1 2094.9 2131.5 1931.9* 207'7.1'1
2349.4' 3013.2 3715.6 4852.5 4931.43 5104.3 6074.5 6231.4 6354.4 6518.9 6973.5. 8192.9 8295.3
11496.5' 12672.4' 12'774.7
lioxide k.*rvrrc m.uenun.lY1 (cni' )
667.3 618.5 660.3 720.5 596.8 647.6 740.8 790.8
1886 2094 2137 1932.5 2076.5 2284.5 2349.3 3609 3716 r-860.6 4903.5 5109 6077 6231 6351 6510 69'76 8193 8293
11496.5 126'72.4 127'74.1
4.679 5.175 I 4.016 ,
2.693. 1
* Used as standards to calculate other wavenumbers. 0 Shown In Ff~g.3~ indicated by & .
Bands bracketjzed together me in~refion~ce.
- h!l -
IJO -
so -
1 (a) 15ybs.M.
x= 0.23 atm.cm.
' ('I t?bEt atmJcm
rl
3 . .
AA
(CL) 1.6/, band,X=S560 atm.cm. (e) 1.49 band
Percentage abeorptions of some bands of CO, at*roomtemperature under low resolution.
0
.oir. 400 6, v ff I I 20 600 i
YK 1000 1200 1400
Vibrational level populations of CO,.
Fig, 5.
me Intensity distribution of a spectral line with pure Lorentz broadening, pure Doppler broadening
for lines of unit integrated intensity, and identical half-widths.
Pig.6.
10 I I
f(x)
1.0 I
0.01. I 0.02 0.1 1.0 X
Fig. 7. The Ladenbsg-Eel&e Curve and :I.L.A. and S.L.A.
Fig. 8. Transmission for t?,e statistical model of a band. It 1s assumed t%at ail qectral ilnes are equally intense and that the Lorer.tz lize sha>e 1s valid. 'n' represents the nun-ber of Uses w~tln rzx3om qacing, but the same mean spacing 'd' in each.
SIATlSTlCAl MODE,
- - - ELSASSER ~00:~ I
I
1
‘i l
1
Lu 0 0001 0001. 001 =x+
f 0’ 10 !O
Fig. 9. Absorption ror the statistical and Elsasser models
of a band, asswdng equally intense Lorentz lines.
- 74 -
Fig. 10. Absorption as a function ofpx = SX/d forp= 0.1. The absorption is shovm for the rancorn super?osltlon of
a single Zlsasser band (Nk); t':ro Xlsasser bards (N=2) where
the intensities sze equal (S,=S,) and viaere one band 1s ten times as Intense as the other (S,=;OS,); the random sugerposltion of five Zlsasser 'oazds (X=5) where the intensities in each band sre ec_ual(S,=S,= . . .=S,); the statistical model where all t'ne intensities are equal.
,‘,c ,- *i/d \s r ,nr 1’w0<42330 CI,P.
.
- 79 -
“ Th&.Ladenburg-Relche ?lot compared with - experimental resuits an& the weak and
strong line -approximations.
- m -
1.0
FREQUENC’f ‘s Cm-‘)
9 400 2300 _ 2200
FREQUENCY (cm-‘)
- 81 -
Fig. 33.
A plot of w-w(n) (- ) versus m (Equ. 5.3).
Be
Curve AlAZ represmts the case for ae=O, curve BlB,, represents the .
case for 0~10.01 Be (typ cal) I and n-l. The vertical line m-0 divides
the P-branch from the R-branch, but the vertical line m-99 divides the
two solutions obtained by Malkmus an6 Thomson (21). It can be seen that
the solution. for m>99 represents the region where the first order
interaction term has become the dominatinp one. Also, we can see that
the solution for m<99 represents both the P- and R-branches. Since the .
integrated line intensity of lines with m>99 is very small this solution
hae (I neeligiblc effect on the final value of the emissivity.
- G2 -
GIHLIOGRAPJ'Y
1.
2.
3.
4. Clatt and Weisskopf, "Theoretical Nuclear Physics", John Wiley and Sons, 1952.
5. D. M. Dennison, Rev. Nod. Phys. 2, 260 (1931)
6. R. C. Ilerman and K. E. Shuler, .J. Chem. Phys. 2, 481 (1954)
7.
6.
9.
10.
11.
R. C. llerman and R. F. Wallis. .J. Chem Phys. 2, 637 (1955)
F!. IWkxm, H. li. Nielsen, t!. II. Shaffer and J. IJagFoner, J. Chem Phys. 2, 40 (1957)
D. M. Dennison, Ref. Mod. Phys. 2, 175 (1940)
G. Flerzbery, "Infrared and Raman Spectra", Van Nostrand, (1962)
E. U. Condon and G. H. Shortley, "The Theory of Atomic Spectra", Cambridge University Press, (1951)
12.
13.
14.
15.
16. M. Zemanskv,
s. s. Penner "Quantitative J%lecular Spectroscopy and Gas Emissivities" Pergamon Press, 1959.
Powell and Craseman, "Ouantum Hechanics", Addison-llesley, 1962
t!eltler, "The Quantum Theory of Radiation", Clarendon Press, Oxford, 1954.
H. Hanson, et al. J. Chem. Phys. 27, 40 (1957)
W. Malkmus .J. Opt. Sot. Am. 2, 951 (1963)
R. Ji. Tourin, J. Opt. Sm. Am. 2, 175 (1961)
"Plan&. Radiation Functions and Electronic.Functions", ‘ Federal Works Ap;ency, IJPA, Nat. Rur. Standards, Washington D.C. (1941)
"I!ent and Thermodynamics", McGraw-Rill, 1957.
- 33 -
17. G. 1:. Plnss, J. Dnt. sot. Am. s, 690 (19%)
1C. R. LadenburT and 0. Reiche, Ann.Phvsii-,. 2, 181 (1913).
19. G. ::. 1'1~s and D. I. Fivel, Astrophys. .J. 117, 275 (1353) -
20. 1:. Il. F.lsasser. Phw. Rev. 2, 126 (1938)
21. V. Plalkms and A. Thomon, .I. Ouant. Snectry. and Rnd. Transfer 2, 17 (19V). Also: Ccnvnir Report ZPh-095, "TnFrared Erissivltp of Diatonic Cases for the Anharmonic Vlbratin: Rotator Ilotlel", Xay, 1961; and Convair Report 7I'h-119, "Snfrared Emissivity of Diatoric Cases with Doppler Lineshaye”, Sept. 1961.
22. C. P. Courtoy, CXL. .I. Phys. 2, hOE (1957)
23. 14. S. Benedict and k. K. Plyler, "Hich-Resolution Spectra of I.vdrocarhon Flames", sat. Bur. Standards Circ. llo. 523, p.P. 57-73, 1954.
24. L. D. Kaplan and D. F. Eo:ers, J. CheF Phys. 2, 876 (1056)
25. U. P. Onpenhcim and Y. nen-Aryah, J. Opt. Sot. Am. 5_2, 344 (1963).
26. W. S. Eenedict, R. C. Herman, C. F.. Noore and S. SilverFan, CZIII. J. Phys. 34, G30 (1956)
27. G. N. Plass, J. opt. sot. Am. 19,821 (1953)
28. C. C. Ferriso, J. Chem. Phys. 2, 1955 (lqfi?).
29. D. E. Burch and D. A. Cryvnak, "Infrared Radiation Emitted bv Hot Gases and its Transmission through Synthetic Atmosphercsh, Aeronutronic Report U-1929,; Oct. 1962.
30. !I. Steinberg: and C:. 0. Davies, J. Chem Phys. 2, 1375 (1961)
31. V. Stull and G. Plass J. Opt. Sot. Am. 50. (1960)
- 84 -
Diaorams were tal.en fron the following papers:
D. PI. Dennison, Ref. 5, Fip. 1 (d)
S. S. Penner, Ref. 1. Fig. 2. /
6. Herzber.q, Ref. 10, Fig. 3. Table 2.
R. 11. Tourin, Ref. 14, Fly. 4.
0. N. Plass, Ref. 17, Fiys. 8, 9, 10.
W. Mallmus, Ref. 13, Fits. 11 to 30.
U. P. Oppenheim and Y. Ben-Aryah, Ref. 25, Figs. 31, 32.
I
D 101492/1/129628 K3 2/88 XL & CL
A.B.C. C.P. Rc. 9&1 December, 1966 J. P. Aodgson
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A.R.C. C.P. NC. 981 December, 1966 J. P. Hodgscn
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A-OFTHE lINFR&m RADlLATIoN A SURVEY OFTHE INFRARED RADIATIOR PROPERTIkS OF CARBON DIOXIDE PROPH(TIBS OF CARBON DIOXIDE
This saper consists of a survey of theoretical end experimental approaches to the prediction of the 1m resclutl~ emissivity of an infrared band of a pclyatdo molecule. Spectral band models and molecular models ars discussed, with carbcm dicxlde being taken as an example of such a molecule.
This paper consists of a survey of theoretical and experimentel approaches to the prediction of the low resolution emissivity of an infrared band of a pclyatcmdc molecule. Spectral band models and molecular models are discussed, with carbon dicxide being taken as an example of such a molecule.
L
A.R.C. C.P. NC. 981 December, 1966 J. P. Hcdgscn
A SURVEY OF THE INFRARED RADIATION PROPERTIES OF CARBON DIOXIDE
This paper consists of a survey of thecretxal and experimental approaches to the prediction of the low resolution emissivity of an xdkared band of a pclyatcmic molecule, Spectral bsnd models and molecular models are discussed mith carbon dicxide being taken as an example of such a molecule.
C.P. No. 981
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